De Bruijn sequences: Difference between revisions

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The sequences are named after the Dutch mathematician Nicolaas Govert de Bruijn.

A note on Dutch capitalization: Nicolaas' last name is de Bruijn, the de isn't normally capitalized unless it's the first word in a sentence. Rosetta Code (more or less by default or by fiat) requires the first word in the task name to be capitalized.

In combinatorial mathematics, a de Bruijn sequence of order n on a size-k alphabet (computer science) A is a cyclic sequence in which every possible length-n string (computer science, formal theory) on A occurs exactly once as a contiguous substring.

Such a sequence is denoted by B(k, n) and has length kⁿ, which is also the number of distinct substrings of length n on A;
de Bruijn sequences are therefore optimally short.

There are:

                         (k!)^{k^(n-1)}   ÷   kⁿ

distinct de Bruijn sequences B(k, n).

Task

For this Rosetta Code task, a de Bruijn sequence is to be generated that can be used to shorten a brute-force attack on a PIN-like code lock that does not have an "enter" key and accepts the last n digits entered.

Note: automated teller machines (ATMs) used to work like this, but their software has been updated to not allow a brute-force attack.

Example

A digital door lock with a 4-digit code would have B (10, 4) solutions, with a length of 10,000 (digits).

Therefore, only at most 10,000 + 3 (as the solutions are cyclic or wrap-around) presses are needed to open the lock.

Trying all 4-digit codes separately would require 4 × 10,000 or 40,000 presses.

Task requirements

Generate a de Bruijn sequence for a 4-digit (decimal) PIN code.

Show the length of the generated de Bruijn sequence.
(There are many possible de Bruijn sequences that solve this task, one solution is shown on the discussion page).
Show the first and last 130 digits of the de Bruijn sequence.

Verify that all four-digit (decimal) 1,000 PIN codes are contained within the de Bruijn sequence.

0000, 0001, 0002, 0003, ... 9996, 9997, 9998, 9999 (note the leading zeros).

Reverse the de Bruijn sequence.
Again, perform the (above) verification test.
Replace the 4,444^th digit with a period (.) in the original de Bruijn sequence.

Perform the verification test (again). There should be four PIN codes missing.

(The last requirement is to ensure that the verification tests performs correctly. The verification processes should list any and all missing PIN codes.)

Show all output here, on this page.

Other tasks related to string operations:

Metrics

Counting

Remove/replace

Anagrams/Derangements/shuffling

Find/Search/Determine

Formatting

Song lyrics/poems/Mad Libs/phrases

Tokenize

Sequences

References

Wikipedia entry: de Bruijn sequence.
MathWorld entry: de Bruijn sequence.
An OEIS entry: A166315 lexicographically earliest binary de Bruijn sequences, B(2,n) --- Not B(10,4), but possibly relevant.

11l

Translation of: D

<lang 11l>V digits = ‘0123456789’

F deBruijn(k, n)

  V alphabet = :digits[0 .< k]
  V a = [Byte(0)] * (k * n)
  [Byte] seq

  F db(Int t, Int p) -> N
     I t > @n
        I @n % p == 0
           @seq.extend(@a[1 .< p + 1])
     E
        @a[t] = @a[t - p]
        @db(t + 1, p)
        V j = @a[t - p] + 1
        L j < @k
           @a[t] = j [&] F'F
           @db(t + 1, t)
           j++

  db(1, 1)
  V buf = ‘’
  L(i) seq
     buf ‘’= alphabet[i]

  R buf‘’buf[0 .< n - 1]

F validate(db)

  V found = [0] * 10'000
  [String] errs

  L(i) 0 .< db.len - 3
     V s = db[i .< i + 4]
     I s.is_digit()
        found[Int(s)]++

  L(i) 10'000
     I found[i] == 0
        errs [+]= ‘    PIN number #04 missing’.format(i)
     E I found[i] > 1
        errs [+]= ‘    PIN number #04 occurs #. times’.format(i, found[i])

  I errs.empty
     print(‘  No errors found’)
  E
     V pl = I errs.len == 1 {‘’} E ‘s’
     print(‘  ’String(errs.len)‘ error’pl‘ found:’)
     L(err) errs
        print(err)

V db = deBruijn(10, 4)

print(‘The length of the de Bruijn sequence is ’db.len) print("\nThe first 130 digits of the de Bruijn sequence are: "db[0.<130]) print("\nThe last 130 digits of the de Bruijn sequence are: "db[(len)-130..])

print("\nValidating the deBruijn sequence:") validate(db)

print("\nValidating the reversed deBruijn sequence:") validate(reversed(db))

db[4443] = ‘.’ print("\nValidating the overlaid deBruijn sequence:") validate(db)</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the deBruijn sequence:
  No errors found

Validating the reversed deBruijn sequence:
  No errors found

Validating the overlaid deBruijn sequence:
  4 errors found:
    PIN number 1459 missing
    PIN number 4591 missing
    PIN number 5814 missing
    PIN number 8145 missing

8080 Assembly

<lang 8080asm>bdos: equ 5 ; BDOS entry point putch: equ 2 ; Write character to console puts: equ 9 ; Write string to console org 100h lhld bdos+1 ; Put stack at highest usable address sphl ;;; Generate de_bruijn(10,4) ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; mvi c,40 ; Zero out a[] xra a lxi d,arr zloop: stax d inx d dcr c jnz zloop lxi h,seq ; H = start of sequence lxi b,0101h ; db(1,1) call db_ lxi d,seq ; Allow wrap-around by appending first 3 digits mvi c,3 wrap: ldax d ; get one of first digits mov m,a ; store after last digit inx d ; advance pointers inx h dcr c ; do this 3 times jnz wrap push h ; store end of data ;;; Print length ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; lxi d,slen ; print "Length: " call pstr lxi d,-seq ; calculate length (-seq+seqEnd) dad d call puthl ; print length call pnl ; print newline ;;; Print first and last 130 digits ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; lxi d,sfrst ; print "First 130: " call pstr lxi h,seq ; print first 130 digits call p130 call pnl ; print newline lxi d,slast ; print "Last 130: " call pstr pop h ; Get end of sequence push h lxi d,-130 ; 130th last digit dad d call p130 ; print last 130 digits call pnl call verify ; verify that all numbers are there ;;; Reverse and verify ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; lxi d,srev ; Print "reversing..." call pstr pop h ; HL = address of last digit dcx h push h ; stack = address of last digit lxi d,seq ; DE = address of first digit call rvrs ; Reverse call verify ; Verify that all numbers are there lxi d,seq ; Then reverse again (restoring it) pop h call rvrs ;;; Replace 4444th digit with '.' and verify ;;;;;;;;;;;;;;;;;;;;;; lxi d,s4444 call pstr mvi a,'.' sta seq+4444 call verify rst 0 ;;; db(t,p); t,p in B,C; end of sequence in HL ;;;;;;;;;;;;;;;;;;;; db_: mov a,b ; if t>n (n=4) cpi 5 ; t >= n+1 jc dbelse mov a,c ; 4%p==0, for p in {1,2,3,4}, is false iff p=3 cpi 3 rz ; stop if p=3, i.e. 4%p<>0 lxi d,arr+1 ; copy P elements to seq forom arr[1..] dbextn: ldax d ; take from a[] mov m,a ; store in sequence inx h ; advance pointers inx d dcr c ; and do this P times jnz dbextn ret dbelse: mov a,b ; t - p sub c mvi d,arr/256 mov e,a ; a[] is page-aligned for easier indexing ldax d ; get a[t-p] mov e,b ; store in a[t] stax d push b ; keep T and P inr b ; db(t+1, p) call db_ pop b ; restore T and P mov a,b ; get a[t-p] sub c mvi d,arr/256 mov e,a ldax d ; j = a[t-p] dbloop: inr a ; j++ cpi 10 ; reached K = 10? rnc ; then stop mvi d,arr/256 mov e,b stax d ; a[t] = j push psw ; keep j push b ; keep t and p mov c,b inr b call db_ ; db(t+1, t) pop b ; restore t and p pop psw ; restore j jmp dbloop ;;; Verify that all numbers are there ;;;;;;;;;;;;;;;;;;;;;;;;;;;;; verify: lxi d,sver ; print "Verifying... " call pstr mvi d,0 ; Zero out the flag array lxi b,10000 lxi h,val vzero: mov m,d inx h dcx b mov a,b ora c jnz vzero lxi h,seq ; Sequence pointer donum: push h ; Store sequence pointer push h ; Push two copies lxi h,0 ; Current 4-digit number mvi c,4 ; Number has 4 digits dgtadd: mov d,h ; HL *= 10 mov e,l dad h dad h dad d dad h xthl ; Get sequence pointer mov a,m ; Get digit inx h ; Advance pointer cpi 10 ; Valid digit? jnc dinval ; If not, go do next 4-digit number xthl ; Back to number mov e,a mvi d,0 dad d ; Add digit to number dcr c ; More digits? jnz dgtadd ; Then get digit lxi d,val ; HL is now the current 4-digit number dad d inr m ; val[HL]++ (we've seen it) dinval: pop h ; Pointer to after last valid digit pop h ; Pointer to start of current number inx h ; Get 4-digit number that starts at next digit mov d,h ; Next pointer in DE mov e,l lxi b,-(seq+10000) ; Are we there yet? dad b mov a,h ora l xchg ; Next pointer back in HL jnz donum ; If not done, do next number. lxi h,val ; Done - get start of validation array mvi b,0 ; B will be set if one is missing vnum: mov a,m ; Have we seen HL-val? ana a jnz vnext ; If so, do the next number push h ; Otherwise, keep current address, lxi d,-val ; Subtract val (to get the number) dad d call puthl ; Print this number as being missing mvi b,1 ; Set B, pop h ; and then restore the address vnext: inx h ; Increment the number push h lxi d,-(val+10000) ; Are we there yet? dad d mov a,h ora l pop h jnz vnum ; If not, check next number. dcr b ; At the end, if B was not set, lxi d,snone ; print "none missing", jnz pstr lxi d,smiss ; otherwise, print "missing" jmp pstr ;;; Subroutines ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;; reverse memory starting at DE and ending at HL rvrs: mov b,m ; Load [HL] ldax d ; Load [DE] mov m,a ; [HL] = old [DE] mov a,b stax d ; [DE] = old [HL] inx d ; Move bottom pointer upwards, dcx h ; Move top pointer downwards, mov a,d ; D<H = not there yet cmp h jc rvrs mov a,e ; E<L = not there yet cmp l jc rvrs ret ;;; print number in HL, saving registers puthl: push h ; save registers push d push b lxi b,nbuf ; number buffer pointer push b ; keep it on the stack dgt: lxi b,-10 lxi d,-1 dgtdiv: inx d ; calculate digit dad b jc dgtdiv mvi a,'0'+10 add l pop h ; get pointer from stack dcx h ; go to previous digit mov m,a ; store digit push h ; put pointer back xchg ; are there any more digits? mov a,h ora l jnz dgt ; if so, calculate next digit pop d ; otherwise, get pointer to first digit jmp pstr_ ; and print the resulting string ;;; print 130 digits from the sequence, starting at HL p130: push h push d push b mvi b,130 ; 130 digits p130l: mov a,m ; get current digit adi '0' ; make ASCII inx h ; advance pointer push b ; save pointer and counter push h mvi c,putch ; print character mov e,a call bdos pop h ; restore pointer and counter pop b dcr b ; one fewer character left jnz p130l ; if characters left, print next jmp rsreg ; otherwise, restore registers and return ;;; print newline pnl: lxi d,snl ;;; print string in DE, saving registers pstr: push h ; store registers push d push b pstr_: mvi c,puts ; print string using CP/M call bdos rsreg: pop b ; restore registers pop d pop h ret snl: db 13,10,'$' slen: db 'Length: $' sfrst: db 'First 130: $' slast: db 'Last 130: $' srev: db 'Reversing...',13,10,'$' s4444: db 'Set seq[4444] to `.`...',13,10,'$' sver: db 'Verifying... $' snone: db 'none ' smiss: db 'missing',13,10,'$' db '00000' ; number output buffer nbuf: db ' $' arr: equ ($/256+1)*256 ; Place to store a[] (page-aligned) val: equ arr+40 ; Place to store validation flags seq: equ val+10000 ; Place to store De Bruijn sequence</lang>

Output:

Length: 10003
First 130: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350
Last 130: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000
Verifying... none missing
Reversing...
Verifying... none missing
Set seq[4444] to `.`...
Verifying... 1459 4591 5914 8145 missing

8086 Assembly

<lang asm>putch: equ 2 ; Print character puts: equ 9 ; Print string cpu 8086 bits 16 section .text org 100h ;;; Calculate de_bruijn(10, 4) ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; xor ax,ax ; zero a[] mov di,arr mov cx,20 ; 20 words = 40 bytes rep stosw mov di,seq ; start of sequence mov dx,0101h ; db(1,1) call db_ mov si,seq ; Add first 3 to end for wrapping mov cx,3 rep movsb lea bp,[di-1] ; Store pointer to last digit in BP ;;; Print length ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; mov ah,puts ; Print "Length:" mov dx,slen int 21h mov ax,di ; Length = end - start sub ax,seq call putax ; Print length ;;; Print first and last 130 characters and verify ;;;;;;;;;;;;;;;; mov ah,puts ; Print "First 130..." mov dx,sfrst int 21h mov si,seq ; print first 130 digits call p130 mov ah,puts ; Print "Last 130..." mov dx,slast int 21h mov si,di ; print last 130 digit sub si,130 call p130 call verify ;;;; Reverse the sequence and verify ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; mov ah,puts ; Print "Reversing..." mov dx,srev int 21h mov si,seq ; SI = first digit in sequence mov di,bp ; DI = last digit in sequence call rvrs ; Reverse call verify ; Verify mov si,seq ; Reverse again, putting it back mov di,bp call rvrs ;;; Set seq[4444] to '.' and verify ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; mov ah,puts ; Print "set seq[4444] to '.'" mov dx,s4444 int 21h mov [seq+4444],byte '.' call verify ; Verify ret ;;; db(t, p); t=dh p=dl, di=seq ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; db_: cmp dh,4 ; t>n? (n=4) jbe .els cmp dl,3 ; for p in {1,2,3,4}, 4%p==0 iff p=3 je .out mov si,arr+1 ; add DL=P bytes from a[1..] to sequence mov cl,dl xor ch,ch rep movsb jmp .out .els: xor bh,bh mov bl,dh sub bl,dl ; t - p mov al,[arr+bx] ; al = a[t-p] mov bl,dh ; t mov [arr+bx],al ; a[t] = al push dx ; keep arguments inc dh ; db(++t,p) call db_ pop dx ; restore arguments mov bl,dh ; al = a[t-p] sub bl,dl mov al,[arr+bx] .loop: inc al ; al++ cmp al,10 ; when al>=k, jae .out ; then stop. mov bl,dh mov [arr+bx],al ; a[t] = j push ax ; keep state push dx mov dl,dh ; db(t+1, t) inc dh call db_ pop dx pop ax jmp .loop .out: ret ;;; Verify that all numbers are there ;;;;;;;;;;;;;;;;;;;;;;;;;;;;; verify: mov ah,puts ; Print "verifying..." mov dx,sver int 21h mov di,val ; Zero validation array mov cx,5000 ; 10000 bytes = 5000 words xor ax,ax rep stosw mov di,val mov si,seq ; Pointer to start of sequence mov cx,6409h ; CH=100 (multiplier), CL=9 (highest digit) .num: mov ax,[si] ; Read first two digits cmp ah,cl ; Check that they are valid ja .inval cmp al,cl ja .inval xchg al,ah ; High digit * 10 + low digit aad mul ch ; Multiply by 100 (to add in next two) mov bx,ax mov ax,[si+2] ; Read last two digits cmp ah,cl ; Check that they are valid ja .inval cmp al,cl ja .inval xchg al,ah ; High digit * 10 + low digit aad add bx,ax ; BX = final 4-digit number inc byte [di+bx] ; Mark this 4-digit number as seen .inval: inc si ; Next digit cmp si,seq+10000 ; Are we at the end yet? jne .num ; If not, do next number mov si,val ; For each number < 10000, check if it's there xor cl,cl ; Will be set if a number is missing .test: lodsb ; Do we have this number? test al,al jnz .tnext ; If so, try next number mov ax,si ; Otherwise, print the missing number sub ax,val call putax mov cl,1 ; And set CL .tnext: cmp si,val+10000 ; Are we at the end yet? jne .test test cl,cl mov dx,smiss ; Print "... missing" jnz .print ; if CL is set mov dx,snone ; or "none missing" otherwise. .print: mov ah,puts int 21h ret ;;; Subroutines ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;; Print number in AX putax: push ax ; Keep registers we're changing push dx push bx push di mov di,numbuf ; Pointer to number buffer mov bx,10 ; Divisor .digit: xor dx,dx ; Divide AX by 10 div bx add dl,'0' ; Add '0' to remainder (digit) dec di ; Store digit in buffer mov [di],dl test ax,ax ; Any more digits? jnz .digit ; If so, do next digits mov dx,di ; At the end, print the string mov ah,puts int 21h pop di ; Restore registers pop bx pop dx pop ax ret ;;; Print 130 digits starting at SI p130: mov cx,130 ; 130 characters mov ah,putch ; Print characters .loop: lodsb ; Get digit add al,'0' ; Make ASCII mov dl,al ; Print digit int 21h loop .loop ret ;;; Reverse memory starting at SI and ending at DI rvrs: mov al,[si] ; Load [SI], mov ah,[di] ; Load [DI], mov [di],al ; Set [DI] = old [SI] mov [si],ah ; Set [SI] = old [DI] inc si ; Increment bottom pointer dec di ; Decrement top pointer cmp si,di ; If SI >= DI, we're done jb rvrs ret section .data slen: db 'Length: $' sfrst: db 13,10,'First 130: $' slast: db 13,10,'Last 130: $' srev: db 13,10,'Reversing... $' s4444: db 13,10,'Set seq[4444] to `.`...$' sver: db 13,10,'Verifying... $' snone: db 'none ' smiss: db 'missing.$' db '00000' numbuf: db ' $' section .bss arr: resb 40 ; a[] val: resb 10000 ; validation array seq: equ $</lang>

Output:

Length: 10003
First 130: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350
Last 130: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000
Verifying... none missing.
Reversing...
Verifying... none missing.
Set seq[4444] to `.`...
Verifying... 1460 4592 5915 8146 missing.

Ada

Translation of: D

<lang Ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Strings.Fixed; use Ada.Strings; with Ada.Strings.Unbounded;

procedure De_Bruijn_Sequences is

  function De_Bruijn (K, N : Positive) return String
  is
     use Ada.Strings.Unbounded;

     Alphabet : constant String := "0123456789";

     subtype Decimal is Integer range 0 .. 9;
     type Decimal_Array is array (Natural range <>) of Decimal;

     A   : Decimal_Array (0 .. K * N - 1) := (others => 0);
     Seq : Unbounded_String;

     procedure Db (T, P : Positive) is
     begin
        if T > N then
           if N mod P = 0 then
              for E of A (1 .. P) loop
                 Append (Seq, Alphabet (Alphabet'First + E));
              end loop;
           end if;
        else
           A (T) := A (T - P);
           Db (T + 1, P);
           for J in A (T - P) + 1 .. K - 1 loop
              A (T) := J;
              Db (T + 1, T);
           end loop;
        end if;
     end Db;

  begin
     Db (1, 1);
     return To_String (Seq) & Slice (Seq, 1, N - 1);
  end De_Bruijn;

  function Image (Value : Integer) return String
  is (Fixed.Trim (Value'Image, Left));

  function PIN_Image (Value : Integer) return String
  is (Fixed.Tail (Image (Value), Count => 4, Pad => '0'));

  procedure Validate (Db : String)
  is
     Found  : array (0 .. 9_999) of Natural := (others => 0);
     Errors : Natural := 0;
  begin

     --  Check all strings of 4 consecutive digits within 'db'
     --  to see if all 10,000 combinations occur without duplication.
     for A in Db'First .. Db'Last - 3 loop
        declare
           PIN : String renames Db (A .. A + 4 - 1);
        begin
           if (for all Char of PIN => Char in '0' .. '9') then
              declare
                 N : constant Integer := Integer'Value (PIN);
                 F : Natural renames Found (N);
              begin
                 F := F + 1;
              end;
           end if;
        end;
     end loop;

     for I in 0_000 .. 9_999 loop
        if Found (I) = 0 then
           Put_Line ("  PIN number " & PIN_Image (I) & " missing");
           Errors := Errors + 1;
        elsif Found (I) > 1 then
           Put_Line ("  PIN number " & PIN_Image (I) & " occurs "
                       & Image (Found (I)) & " times");
           Errors := Errors + 1;
        end if;
     end loop;

     case Errors is
        when 0 =>  Put_Line ("  No errors found");
        when 1 =>  Put_Line ("  1 error found");
        when others =>
           Put_Line ("  " & Image (Errors) & " errors found");
     end case;
  end Validate;

  function Backwards (S : String) return String is
     R : String (S'Range);
  begin
     for A in 0 .. S'Length - 1 loop
        R (R'Last - A) := S (S'First + A);
     end loop;
     return R;
  end Backwards;

  DB  : constant String := De_Bruijn (K => 10, N => 4);
  Rev : constant String := Backwards (DB);
  Ovl :          String := DB;

begin

  Put_Line ("The length of the de Bruijn sequence is " & DB'Length'Image);
  New_Line;

  Put_Line ("The first 130 digits of the de Bruijn sequence are: ");
  Put_Line ("  " & Fixed.Head (DB, 130));
  New_Line;

  Put_Line ("The last 130 digits of the de Bruijn sequence are: ");
  Put_Line ("  " & Fixed.Tail (DB, 130));
  New_Line;

  Put_Line ("Validating the deBruijn sequence:");
  Validate (DB);
  New_Line;

  Put_Line ("Validating the reversed deBruijn sequence:");
  Validate (Rev);
  New_Line;

  Ovl (4444) := '.';
  Put_Line ("Validating the overlaid deBruijn sequence:");
  Validate (Ovl);
  New_Line;

end De_Bruijn_Sequences;</lang>

Output:

The length of the de Bruijn sequence is  10003

The first 130 digits of the de Bruijn sequence are:
  0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are:
  6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the deBruijn sequence:
  No errors found

Validating the reversed deBruijn sequence:
  No errors found

Validating the overlaid deBruijn sequence:
  PIN number 1459 missing
  PIN number 4591 missing
  PIN number 5814 missing
  PIN number 8145 missing
  4 errors found

BASIC

<lang gwbasic>10 DEFINT A-Z 20 K = 10: N = 4 30 DIM A(K*N), S(K^N+N), T(5), P(5), V(K^N\8) 40 GOSUB 200 50 PRINT "Length: ",S 60 PRINT "First 130:" 70 FOR I=0 TO 129: PRINT USING "#";S(I);: NEXT 80 PRINT: PRINT "Last 130:" 90 FOR I=S-130 TO S-1: PRINT USING "#";S(I);: NEXT 100 PRINT 110 GOSUB 600 120 PRINT "Reversing...": GOSUB 500: GOSUB 600: GOSUB 500 130 PRINT USING "Replacing 4444'th element (#):";S(4443) 140 S(4443) = -1 : REM 0-indexed, and using integers 150 GOSUB 600 160 END 200 REM Generate De Bruijn sequence given K and N 210 T(R) = 1: P(R) = 1 220 IF T(R) > N GOTO 380 230 A(T(R)) = A(T(R)-P(R)) 240 R = R+1 250 T(R) = T(R-1)+1 260 P(R) = P(R-1) 270 GOSUB 220 280 R = R-1 290 FOR J = A(T(R)-P(R))+1 TO K-1 300 A(T(R)) = J 310 R = R+1 320 T(R) = T(R-1)+1 330 P(R) = T(R-1) 340 GOSUB 220 350 R = R-1 355 J = A(T(R)) 360 NEXT 370 RETURN 380 IF N MOD P(R) THEN RETURN 390 FOR I = 1 TO P(R) 400 S(S) = A(I) 410 S = S+1 420 NEXT 430 RETURN 500 REM Reverse the sequence 510 FOR I=0 TO S\2 520 J = S(I) 530 S(I) = S(S-I) 540 S(S-I) = J 550 NEXT 560 RETURN 600 REM Validate the sequence (uses bit packing to save memory) 610 PRINT "Validating..."; 620 FOR I=0 TO N-1: S(S+I)=S(I): NEXT 630 FOR I=0 TO K^N\8-1: V(I)=0: NEXT 640 FOR I=0 TO S 650 P=0 660 FOR J=0 TO N-1 662 D=S(I+J) 663 IF D<0 GOTO 690 665 P=K*P+D 669 NEXT J 670 X=P\8 680 V(X) = V(X) OR 2^(P AND 7) 690 NEXT I 700 M=1 710 FOR I=0 TO K^N\8-1 720 IF V(I)=255 GOTO 760 730 FOR J=0 TO 7 740 IF (V(I) AND 2^J)=0 THEN M=0: PRINT USING " ####";I*8+J; 750 NEXT 760 NEXT 770 IF M THEN PRINT " none"; 780 PRINT " missing." 790 RETURN</lang>

Output:

Length:        10000
First 130:
00001000200030004000500060007000800090011001200130014001500160017001800190021002
20023002400250026002700280029003100320033003400350
Last 130:
89768986899696977697869796987698869896997699869997777877797788778977987799787879
78887889789878997979887989799879998888988998989999
Validating... none missing.
Reversing...
Validating... none missing.
Replacing 4444'th element (4):
Validating... 1459 4591 5814 8145 missing.

C#

Translation of: Kotlin

<lang csharp>using System; using System.Collections.Generic; using System.Text;

namespace DeBruijn {

   class Program {
       const string digits = "0123456789";

       static string DeBruijn(int k, int n) {
           var alphabet = digits.Substring(0, k);
           var a = new byte[k * n];
           var seq = new List<byte>();
           void db(int t, int p) {
               if (t > n) {
                   if (n % p == 0) {
                       seq.AddRange(new ArraySegment<byte>(a, 1, p));
                   }
               } else {
                   a[t] = a[t - p];
                   db(t + 1, p);
                   var j = a[t - p] + 1;
                   while (j < k) {
                       a[t] = (byte)j;
                       db(t + 1, t);
                       j++;
                   }
               }
           }
           db(1, 1);
           var buf = new StringBuilder();
           foreach (var i in seq) {
               buf.Append(alphabet[i]);
           }
           var b = buf.ToString();
           return b + b.Substring(0, n - 1);
       }

       static bool AllDigits(string s) {
           foreach (var c in s) {
               if (c < '0' || '9' < c) {
                   return false;
               }
           }
           return true;
       }

       static void Validate(string db) {
           var le = db.Length;
           var found = new int[10_000];
           var errs = new List<string>();
           // Check all strings of 4 consecutive digits within 'db'
           // to see if all 10,000 combinations occur without duplication.
           for (int i = 0; i < le - 3; i++) {
               var s = db.Substring(i, 4);
               if (AllDigits(s)) {
                   int.TryParse(s, out int n);
                   found[n]++;
               }
           }
           for (int i = 0; i < 10_000; i++) {
               if (found[i] == 0) {
                   errs.Add(string.Format("    PIN number {0,4} missing", i));
               } else if (found[i] > 1) {
                   errs.Add(string.Format("    PIN number {0,4} occurs {1} times", i, found[i]));
               }
           }
           var lerr = errs.Count;
           if (lerr == 0) {
               Console.WriteLine("  No errors found");
           } else {
               var pl = lerr == 1 ? "" : "s";
               Console.WriteLine("  {0} error{1} found:", lerr, pl);
               errs.ForEach(Console.WriteLine);
           }
       }

       static string Reverse(string s) {
           char[] arr = s.ToCharArray();
           Array.Reverse(arr);
           return new string(arr);
       }

       static void Main() {
           var db = DeBruijn(10, 4);
           var le = db.Length;

           Console.WriteLine("The length of the de Bruijn sequence is {0}", le);
           Console.WriteLine("\nThe first 130 digits of the de Bruijn sequence are: {0}", db.Substring(0, 130));
           Console.WriteLine("\nThe last 130 digits of the de Bruijn sequence are: {0}", db.Substring(le - 130, 130));

           Console.WriteLine("\nValidating the deBruijn sequence:");
           Validate(db);

           Console.WriteLine("\nValidating the reversed deBruijn sequence:");
           Validate(Reverse(db));

           var bytes = db.ToCharArray();
           bytes[4443] = '.';
           db = new string(bytes);
           Console.WriteLine("\nValidating the overlaid deBruijn sequence:");
           Validate(db);
       }
   }

}</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the deBruijn sequence:
  No errors found

Validating the reversed deBruijn sequence:
  No errors found

Validating the overlaid deBruijn sequence:
  4 errors found:
    PIN number 1459 missing
    PIN number 4591 missing
    PIN number 5814 missing
    PIN number 8145 missing

C++

Translation of: D

<lang cpp>#include <algorithm>

include <functional>
include <iostream>
include <iterator>
include <string>
include <sstream>
include <vector>

typedef unsigned char byte;

std::string deBruijn(int k, int n) {

   std::vector<byte> a(k * n, 0);
   std::vector<byte> seq;

   std::function<void(int, int)> db;
   db = [&](int t, int p) {
       if (t > n) {
           if (n % p == 0) {
               for (int i = 1; i < p + 1; i++) {
                   seq.push_back(a[i]);
               }
           }
       } else {
           a[t] = a[t - p];
           db(t + 1, p);
           auto j = a[t - p] + 1;
           while (j < k) {
               a[t] = j & 0xFF;
               db(t + 1, t);
               j++;
           }
       }
   };

   db(1, 1);
   std::string buf;
   for (auto i : seq) {
       buf.push_back('0' + i);
   }
   return buf + buf.substr(0, n - 1);

}

bool allDigits(std::string s) {

   for (auto c : s) {
       if (c < '0' || '9' < c) {
           return false;
       }
   }
   return true;

}

void validate(std::string db) {

   auto le = db.size();
   std::vector<int> found(10000, 0);
   std::vector<std::string> errs;

   // Check all strings of 4 consecutive digits within 'db'
   // to see if all 10,000 combinations occur without duplication.
   for (size_t i = 0; i < le - 3; i++) {
       auto s = db.substr(i, 4);
       if (allDigits(s)) {
           auto n = stoi(s);
           found[n]++;
       }
   }

   for (int i = 0; i < 10000; i++) {
       if (found[i] == 0) {
           std::stringstream ss;
           ss << "    PIN number " << i << " missing";
           errs.push_back(ss.str());
       } else if (found[i] > 1) {
           std::stringstream ss;
           ss << "    PIN number " << i << " occurs " << found[i] << " times";
           errs.push_back(ss.str());
       }
   }

   if (errs.empty()) {
       std::cout << "  No errors found\n";
   } else {
       auto pl = (errs.size() == 1) ? "" : "s";
       std::cout << "  " << errs.size() << " error" << pl << " found:\n";
       for (auto e : errs) {
           std::cout << e << '\n';
       }
   }

}

int main() {

   std::ostream_iterator<byte> oi(std::cout, "");
   auto db = deBruijn(10, 4);

   std::cout << "The length of the de Bruijn sequence is " << db.size() << "\n\n";
   std::cout << "The first 130 digits of the de Bruijn sequence are: ";
   std::copy_n(db.cbegin(), 130, oi);
   std::cout << "\n\nThe last 130 digits of the de Bruijn sequence are: ";
   std::copy(db.cbegin() + (db.size() - 130), db.cend(), oi);
   std::cout << "\n";

   std::cout << "\nValidating the de Bruijn sequence:\n";
   validate(db);

   std::cout << "\nValidating the reversed de Bruijn sequence:\n";
   auto rdb = db;
   std::reverse(rdb.begin(), rdb.end());
   validate(rdb);

   auto by = db;
   by[4443] = '.';
   std::cout << "\nValidating the overlaid de Bruijn sequence:\n";
   validate(by);

   return 0;

}</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the de Bruijn sequence:
  No errors found

Validating the reversed de Bruijn sequence:
  No errors found

Validating the overlaid de Bruijn sequence:
  4 errors found:
    PIN number 1459 missing
    PIN number 4591 missing
    PIN number 5814 missing
    PIN number 8145 missing

CLU

<lang clu>% Generate the De Bruijn sequence consisiting of N-digit numbers de_bruijn = cluster is generate

   rep = null
   own k: int := 0
   own n: int := 0
   own a: array[int] := array[int]$[]
   own seq: array[int] := array[int]$[]
   
   generate = proc (k_, n_: int) returns (string)
       k := k_
       n := n_
       a := array[int]$fill(0, k*n, 0)
       seq := array[int]$[]
       db(1, 1)
       s: stream := stream$create_output()
       for i: int in array[int]$elements(seq) do
           stream$puts(s, int$unparse(i))
       end
       return(stream$get_contents(s))
   end generate
   
   db = proc (t, p: int)
       if t>n then 
           if n//p = 0 then
               for i: int in int$from_to(1, p) do
                   array[int]$addh(seq, a[i])
               end
           end
       else
           a[t] := a[t - p]
           db(t+1, p)
           for j: int in int$from_to(a[t - p] + 1, k-1) do
               a[t] := j
               db(t + 1, t)
           end
       end
   end db

end de_bruijn

% Reverse a string reverse = proc (s: string) returns (string)

   r: array[char] := array[char]$predict(1, string$size(s))
   for c: char in string$chars(s) do
       array[char]$addl(r, c)
   end
   return(string$ac2s(r))

end reverse

% Find all missing N-digit values find_missing = proc (db: string, n: int) returns (sequence[string])

   db := db || string$substr(db, 1, n) % wrap
   missing: array[string] := array[string]$[]
   s: stream := stream$create_output()
   for i: int in int$from_to(0, 10**n-1) do
       %s: stream := stream$create_output()
       stream$reset(s)
       stream$putzero(s, int$unparse(i), n)
       val: string := stream$get_contents(s)
       if string$indexs(val, db) = 0 then
           array[string]$addh(missing, val)
       end
   end
   return(sequence[string]$a2s(missing))

end find_missing

% Report all missing values, or 'none'. validate = proc (s: stream, db: string, n: int)

   stream$puts(s, "Validating...")
   missing: sequence[string] := find_missing(db, n)
   for v: string in sequence[string]$elements(missing) do
       stream$puts(s, " " || v)
   end
   if sequence[string]$size(missing) = 0 then
       stream$puts(s, " none")
   end
   stream$putl(s, " missing.")

end validate

start_up = proc ()

   po: stream := stream$primary_output()
   
   % Generate the De Bruijn sequence for 4-digit numbers
   db: string := de_bruijn$generate(10, 4)
   
   % Report length and first and last digits
   stream$putl(po, "Length: " || int$unparse(string$size(db)))
   stream$putl(po, "First 130 characters:")
   stream$putl(po, string$substr(db, 1, 130))
   stream$putl(po, "Last 130 characters:")
   stream$putl(po, string$substr(db, string$size(db)-130, 130))
   
   % See if there are any missing values in the sequence
   validate(po, db, 4)
   
   % Reverse and validate again
   stream$putl(po, "Reversing...")
   validate(po, reverse(db), 4)
   
   % Replace the 4444'th element with '.' and validate again
   stream$putl(po, "Setting the 4444'th character to '.'...")
   db := string$substr(db, 1, 4443) || "." || string$rest(db, 4445)
   validate(po, db, 4)

end start_up</lang>

Output:

Length: 10000
First 130 characters:
0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350
Last 130 characters:
6897689868996969776978697969876988698969976998699977778777977887789779877997878797888788978987899797988798979987999888898899898999
Validating... none missing.
Reversing...
Validating... none missing.
Setting the 4444'th character to '.'...
Validating... 1459 4591 5814 8145 missing.

D

Translation of: Kotlin

<lang d>import std.array; import std.conv; import std.format; import std.range; import std.stdio;

immutable DIGITS = "0123456789";

string deBruijn(int k, int n) {

   auto alphabet = DIGITS[0..k];
   byte[] a;
   a.length = k * n;
   byte[] seq;

   void db(int t, int p) {
       if (t > n) {
           if (n % p == 0) {
               auto temp = a[1..p + 1];
               seq ~= temp;
           }
       } else {
           a[t] = a[t - p];
           db(t + 1, p);
           auto j = a[t - p] + 1;
           while (j < k) {
               a[t] = cast(byte)(j & 0xFF);
               db(t + 1, t);
               j++;
           }
       }
   }
   db(1, 1);
   string buf;
   foreach (i; seq) {
       buf ~= alphabet[i];
   }

   return buf ~ buf[0 .. n - 1];

}

bool allDigits(string s) {

   foreach (c; s) {
       if (c < '0' || '9' < c) {
           return false;
       }
   }
   return true;

}

void validate(string db) {

   auto le = db.length;
   int[10_000] found;
   string[] errs;
   // Check all strings of 4 consecutive digits within 'db'
   // to see if all 10,000 combinations occur without duplication.
   foreach (i; 0 .. le - 3) {
       auto s = db[i .. i + 4];
       if (allDigits(s)) {
           auto n = s.to!int;
           found[n]++;
       }
   }
   foreach (i; 0 .. 10_000) {
       if (found[i] == 0) {
           errs ~= format("    PIN number %04d missing", i);
       } else if (found[i] > 1) {
           errs ~= format("    PIN number %04d occurs %d times", i, found[i]);
       }
   }
   if (errs.empty) {
       writeln("  No errors found");
   } else {
       auto pl = (errs.length == 1) ? "" : "s";
       writeln("  ", errs.length, " error", pl, " found:");
       writefln("%-(%s\n%)", errs);
   }

}

void main() {

   auto db = deBruijn(10, 4);

   writeln("The length of the de Bruijn sequence is ", db.length);
   writeln("\nThe first 130 digits of the de Bruijn sequence are: ", db[0 .. 130]);
   writeln("\nThe last 130 digits of the de Bruijn sequence are: ", db[$ - 130 .. $]);

   writeln("\nValidating the deBruijn sequence:");
   validate(db);

   writeln("\nValidating the reversed deBruijn sequence:");
   validate(db.retro.to!string);

   auto by = db.dup;
   by[4443] = '.';
   db = by.idup;
   writeln("\nValidating the overlaid deBruijn sequence:");
   validate(db);

}</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the deBruijn sequence:
  No errors found

Validating the reversed deBruijn sequence:
  No errors found

Validating the overlaid deBruijn sequence:
  4 errors found:
    PIN number 1459 missing
    PIN number 4591 missing
    PIN number 5814 missing
    PIN number 8145 missing

Go

<lang go>package main

import (

   "bytes"
   "fmt"
   "strconv"
   "strings"

)

const digits = "0123456789"

func deBruijn(k, n int) string {

   alphabet := digits[0:k]
   a := make([]byte, k*n)
   var seq []byte
   var db func(int, int) // recursive closure
   db = func(t, p int) {
       if t > n {
           if n%p == 0 {
               seq = append(seq, a[1:p+1]...)
           }
       } else {
           a[t] = a[t-p]
           db(t+1, p)
           for j := int(a[t-p] + 1); j < k; j++ {
               a[t] = byte(j)
               db(t+1, t)
           }
       }
   }
   db(1, 1)
   var buf bytes.Buffer
   for _, i := range seq {
       buf.WriteByte(alphabet[i])
   }
   b := buf.String()
   return b + b[0:n-1] // as cyclic append first (n-1) digits

}

func allDigits(s string) bool {

   for _, b := range s {
       if b < '0' || b > '9' {
           return false
       }
   }
   return true

}

func validate(db string) {

   le := len(db)
   found := make([]int, 10000)
   var errs []string
   // Check all strings of 4 consecutive digits within 'db'
   // to see if all 10,000 combinations occur without duplication.
   for i := 0; i < le-3; i++ {
       s := db[i : i+4]
       if allDigits(s) {
           n, _ := strconv.Atoi(s)
           found[n]++
       }
   }
   for i := 0; i < 10000; i++ {
       if found[i] == 0 {
           errs = append(errs, fmt.Sprintf("    PIN number %04d missing", i))
       } else if found[i] > 1 {
           errs = append(errs, fmt.Sprintf("    PIN number %04d occurs %d times", i, found[i]))
       }
   }
   lerr := len(errs)
   if lerr == 0 {
       fmt.Println("  No errors found")
   } else {
       pl := "s"
       if lerr == 1 {
           pl = ""
       }
       fmt.Printf("  %d error%s found:\n", lerr, pl)
       fmt.Println(strings.Join(errs, "\n"))
   }

}

func reverse(s string) string {

   bytes := []byte(s)
   for i, j := 0, len(s)-1; i < j; i, j = i+1, j-1 {
       bytes[i], bytes[j] = bytes[j], bytes[i]
   }
   return string(bytes)

}

func main() {

   db := deBruijn(10, 4)
   le := len(db)
   fmt.Println("The length of the de Bruijn sequence is", le)
   fmt.Println("\nThe first 130 digits of the de Bruijn sequence are:")
   fmt.Println(db[0:130])
   fmt.Println("\nThe last 130 digits of the de Bruijn sequence are:")
   fmt.Println(db[le-130:])
   fmt.Println("\nValidating the de Bruijn sequence:")
   validate(db)

   fmt.Println("\nValidating the reversed de Bruijn sequence:")
   dbr := reverse(db)
   validate(dbr)

   bytes := []byte(db)
   bytes[4443] = '.'
   db = string(bytes)
   fmt.Println("\nValidating the overlaid de Bruijn sequence:")
   validate(db)

}</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are:
0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are:
6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the de Bruijn sequence:
  No errors found

Validating the reversed de Bruijn sequence:
  No errors found

Validating the overlaid de Bruijn sequence:
  4 errors found:
    PIN number 1459 missing
    PIN number 4591 missing
    PIN number 5814 missing
    PIN number 8145 missing

Groovy

Translation of: Java

<lang groovy>import java.util.function.BiConsumer

class DeBruijn {

   interface Recursable<T, U> {
       void apply(T t, U u, Recursable<T, U> r);
   }

   static <T, U> BiConsumer<T, U> recurse(Recursable<T, U> f) {
       return { t, u -> f.apply(t, u, f) }
   }

   private static String deBruijn(int k, int n) {
       byte[] a = new byte[k * n]
       Arrays.fill(a, (byte) 0)

       List<Byte> seq = new ArrayList<>()

       BiConsumer<Integer, Integer> db = recurse({ int t, int p, f ->
           if (t > n) {
               if (n % p == 0) {
                   for (int i = 1; i < p + 1; ++i) {
                       seq.add(a[i])
                   }
               }
           } else {
               a[t] = a[t - p]
               f.apply(t + 1, p, f)
               int j = a[t - p] + 1
               while (j < k) {
                   a[t] = (byte) (j & 0xFF)
                   f.apply(t + 1, t, f)
                   j++
               }
           }
       })
       db.accept(1, 1)

       StringBuilder sb = new StringBuilder()
       for (Byte i : seq) {
           sb.append("0123456789".charAt(i))
       }

       sb.append(sb.subSequence(0, n - 1))
       return sb.toString()
   }

   private static boolean allDigits(String s) {
       for (int i = 0; i < s.length(); ++i) {
           char c = s.charAt(i)
           if (!Character.isDigit(c)) {
               return false
           }
       }
       return true
   }

   private static void validate(String db) {
       int le = db.length()
       int[] found = new int[10_000]
       Arrays.fill(found, 0)
       List<String> errs = new ArrayList<>()

       // Check all strings of 4 consecutive digits within 'db'
       // to see if all 10,000 combinations occur without duplication.
       for (int i = 0; i < le - 3; ++i) {
           String s = db.substring(i, i + 4)
           if (allDigits(s)) {
               int n = Integer.parseInt(s)
               found[n]++
           }
       }

       for (int i = 0; i < 10_000; ++i) {
           if (found[i] == 0) {
               errs.add(String.format("    PIN number %d is missing", i))
           } else if (found[i] > 1) {
               errs.add(String.format("    PIN number %d occurs %d times", i, found[i]))
           }
       }

       if (errs.isEmpty()) {
           System.out.println("    No errors found")
       } else {
           String pl = (errs.size() == 1) ? "" : "s"
           System.out.printf("  %d error%s found:\n", errs.size(), pl)
           errs.forEach(System.out.&println)
       }
   }

   static void main(String[] args) {
       String db = deBruijn(10, 4)

       System.out.printf("The length of the de Bruijn sequence is %d\n\n", db.length())
       System.out.printf("The first 130 digits of the de Bruijn sequence are: %s\n\n", db.substring(0, 130))
       System.out.printf("The last 130 digits of the de Bruijn sequence are: %s\n\n", db.substring(db.length() - 130))

       System.out.println("Validating the de Bruijn sequence:")
       validate(db)

       StringBuilder sb = new StringBuilder(db)
       String rdb = sb.reverse().toString()
       System.out.println()
       System.out.println("Validating the de Bruijn sequence:")
       validate(rdb)

       sb = new StringBuilder(db)
       sb.setCharAt(4443, '.' as char)
       System.out.println()
       System.out.println("Validating the overlaid de Bruijn sequence:")
       validate(sb.toString())
   }

}</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the de Bruijn sequence:
    No errors found

Validating the de Bruijn sequence:
    No errors found

Validating the overlaid de Bruijn sequence:
  4 errors found:
    PIN number 1459 is missing
    PIN number 4591 is missing
    PIN number 5814 is missing
    PIN number 8145 is missing

Haskell

Permutation-based

Straight-forward implementation of inverse Burrows—Wheeler transform [1] is reasonably efficient for the task (about a milliseconds for B(10,4) in GHCi).

<lang haskell>import Data.List import Data.Map ((!)) import qualified Data.Map as M

-- represents a permutation in a cycle notation cycleForm :: [Int] -> Int cycleForm p = unfoldr getCycle $ M.fromList $ zip [0..] p

 where
   getCycle p
     | M.null p = Nothing
     | otherwise = 
         let Just ((x,y), m) = M.minViewWithKey p
             c = if x == y then [] else takeWhile (/= x) (iterate (m !) y)
         in Just (c ++ [x], foldr M.delete m c)

-- the set of Lyndon words generated by inverse Burrows—Wheeler transform lyndonWords :: Ord a => [a] -> Int -> a lyndonWords s n = map (ref !!) <$> cycleForm perm

 where
   ref = concat $ replicate (length s ^ (n - 1)) s
   perm = s >>= (`elemIndices` ref)

-- returns the de Bruijn sequence of order n for an alphabeth s deBruijn :: Ord a => [a] -> Int -> [a] deBruijn s n = let lw = concat $ lyndonWords n s

              in lw ++ take (n-1) lw</lang>

λ> cycleForm [1,4,3,2,0]
[[1,4,0],[3,2]]

λ> lyndonWords "ab" 3
["a","aab","abb","b"]

λ> deBruijn "ab" 3
"aaababbbaa"

The task.

<lang haskell>import Control.Monad (replicateM)

main = do

 let symbols = ['0'..'9']
 let db = deBruijn symbols 4
 putStrLn $ "The length of de Bruijn sequence: " ++ show (length db)
 putStrLn $ "The first 130 symbols are:\n" ++ show (take 130 db)
 putStrLn $ "The last 130 symbols are:\n" ++ show (drop (length db - 130) db)

 let words = replicateM 4 symbols
 let validate db  = filter (not . (`isInfixOf` db)) words
 putStrLn $ "Words not in the sequence: " ++ unwords (validate db)

 let db' = a ++ ('.': tail b) where (a,b) = splitAt 4444 db
 putStrLn $ "Words not in the corrupted sequence: " ++ unwords (validate db') </lang>

λ> main
The length of de Bruijn sequence: 10003
The first 130 symbols are:
"0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350"
The last 130 symbols are:
"6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000"
Words not in the sequence: 
Words not in the corrupted sequence: 1459 4591 5914 8145

Array-based

Translation of: Python

<lang haskell>import Control.Monad.State import Data.Array (Array, listArray, (!), (//)) import qualified Data.Array as A

deBruijn :: [a] -> Int -> [a] deBruijn s n =

 let
   k = length s
   
   db :: Int -> Int -> State (Array Int Int) [Int]    
   db t p =
     if t > n
     then 
       if n `mod` p == 0
       then get >>= \a -> return [ a ! k | k <- [1 .. p]]
       else return []
     else do
       a <- get
       x <- setArray t (a ! (t-p)) >> db (t+1) p
       a <- get
       y <- sequence [ setArray t j >> db (t+1) t
                     | j <- [a ! (t-p) + 1 .. k - 1] ]
       return $ x ++ concat y
       
   setArray i x = modify (// [(i, x)])
   
   seqn = db 1 1 `evalState` listArray (0, k*n-1) (repeat 0)
   
 in [ s !! i | i <- seqn ++ take (n-1) seqn ]</lang>

J

definitions. The C. verb computes the cycles. J's sort is not a stable sort. <lang J>NB. implement inverse Burrows—Wheeler transform sequence method

repeat_alphabet=: [: , [: i.&> (^ <:) # [ assert 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 -: 2 repeat_alphabet 4

de_bruijn=: ({~ ([: ; [: C. /:^:2))@:repeat_alphabet NB. K de_bruijn N

pins=: #&10 #: [: i. 10&^ NB. pins y generates all y digit PINs groups=: [ ]\ ] , ({.~ <:)~ NB. length x infixes of sequence y cyclically extended by x-1 verify_PINs=: (/:~@:groups -: pins@:[) NB. LENGTH verify_PINs SEQUENCE </lang>Task<lang J> NB. A is the sequence

  A=: 10 de_bruijn 4

  NB. tally A
  #A

10000

  NB. literally the first and final 130 digits
  Num_j_ {~ 130 ({. ,: ({.~ -)~) A

0000101001101111000210020102110202001210120112111202121200221022012211220222122220003100320030103110321030203120322030300131013201 9469956996699769986990799179927993799479957996799779987990899189928993899489958996899789988990999199929993999499959996999799989999

  NB. verifications.  seriously?
  4 verify_PINs A

1

  4 (verify_PINs |.) A

1

  4 verify_PINs (a.i.'.') (<: 4444)} A

0 </lang>

Java

Translation of: C++

<lang java>import java.util.ArrayList; import java.util.Arrays; import java.util.List; import java.util.function.BiConsumer;

public class DeBruijn {

   public interface Recursable<T, U> {
       void apply(T t, U u, Recursable<T, U> r);
   }

   public static <T, U> BiConsumer<T, U> recurse(Recursable<T, U> f) {
       return (t, u) -> f.apply(t, u, f);
   }

   private static String deBruijn(int k, int n) {
       byte[] a = new byte[k * n];
       Arrays.fill(a, (byte) 0);

       List<Byte> seq = new ArrayList<>();

       BiConsumer<Integer, Integer> db = recurse((t, p, f) -> {
           if (t > n) {
               if (n % p == 0) {
                   for (int i = 1; i < p + 1; ++i) {
                       seq.add(a[i]);
                   }
               }
           } else {
               a[t] = a[t - p];
               f.apply(t + 1, p, f);
               int j = a[t - p] + 1;
               while (j < k) {
                   a[t] = (byte) (j & 0xFF);
                   f.apply(t + 1, t, f);
                   j++;
               }
           }
       });
       db.accept(1, 1);

       StringBuilder sb = new StringBuilder();
       for (Byte i : seq) {
           sb.append("0123456789".charAt(i));
       }

       sb.append(sb.subSequence(0, n - 1));
       return sb.toString();
   }

   private static boolean allDigits(String s) {
       for (int i = 0; i < s.length(); ++i) {
           char c = s.charAt(i);
           if (!Character.isDigit(c)) {
               return false;
           }
       }
       return true;
   }

   private static void validate(String db) {
       int le = db.length();
       int[] found = new int[10_000];
       Arrays.fill(found, 0);
       List<String> errs = new ArrayList<>();

       // Check all strings of 4 consecutive digits within 'db'
       // to see if all 10,000 combinations occur without duplication.
       for (int i = 0; i < le - 3; ++i) {
           String s = db.substring(i, i + 4);
           if (allDigits(s)) {
               int n = Integer.parseInt(s);
               found[n]++;
           }
       }

       for (int i = 0; i < 10_000; ++i) {
           if (found[i] == 0) {
               errs.add(String.format("    PIN number %d is missing", i));
           } else if (found[i] > 1) {
               errs.add(String.format("    PIN number %d occurs %d times", i, found[i]));
           }
       }

       if (errs.isEmpty()) {
           System.out.println("    No errors found");
       } else {
           String pl = (errs.size() == 1) ? "" : "s";
           System.out.printf("  %d error%s found:\n", errs.size(), pl);
           errs.forEach(System.out::println);
       }
   }

   public static void main(String[] args) {
       String db = deBruijn(10, 4);

       System.out.printf("The length of the de Bruijn sequence is %d\n\n", db.length());
       System.out.printf("The first 130 digits of the de Bruijn sequence are: %s\n\n", db.substring(0, 130));
       System.out.printf("The last 130 digits of the de Bruijn sequence are: %s\n\n", db.substring(db.length() - 130));

       System.out.println("Validating the de Bruijn sequence:");
       validate(db);

       StringBuilder sb = new StringBuilder(db);
       String rdb = sb.reverse().toString();
       System.out.println();
       System.out.println("Validating the de Bruijn sequence:");
       validate(rdb);

       sb = new StringBuilder(db);
       sb.setCharAt(4443, '.');
       System.out.println();
       System.out.println("Validating the overlaid de Bruijn sequence:");
       validate(sb.toString());
   }

}</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the de Bruijn sequence:
    No errors found

Validating the de Bruijn sequence:
    No errors found

Validating the overlaid de Bruijn sequence:
  4 errors found:
    PIN number 1459 is missing
    PIN number 4591 is missing
    PIN number 5814 is missing
    PIN number 8145 is missing

Julia

<lang julia>function debruijn(k::Integer, n::Integer)

   alphabet = b"0123456789abcdefghijklmnopqrstuvwxyz"[1:k]
   a = zeros(UInt8, k * n)
   seq = UInt8[]

   function db(t, p)
       if t > n
           if n % p == 0
               append!(seq, a[2:p+1])
           end
       else
           a[t + 1] = a[t - p + 1]
           db(t + 1, p)
           for j in a[t-p+1]+1:k-1
               a[t + 1] = j
               db(t + 1, t)
           end
       end
   end

   db(1, 1)
   return String([alphabet[i + 1] for i in vcat(seq, seq[1:n-1])])

end

function verifyallPIN(str, k, n, deltaposition=0)

   if deltaposition != 0
       str = str[1:deltaposition-1] * "." * str[deltaposition+1:end]
   end
   result = true
   for i in 1:k^n-1
       pin = string(i, pad=n)
       if !occursin(pin, str)
           println("PIN $pin does not occur in the sequence.")
           result = false
       end
   end
   println("The sequence does ", result ? "" : "not ", "contain all PINs.")

end

const s = debruijn(10, 4) println("The length of the sequence is $(length(s)). The first 130 digits are:\n",

   s[1:130], "\nand the last 130 digits are:\n", s[end-130:end])

print("Testing sequence: "), verifyallPIN(s, 10, 4) print("Testing the reversed sequence: "), verifyallPIN(reverse(s), 10, 4) println("\nAfter replacing 4444th digit with \'.\':"), verifyallPIN(s, 10, 4, 4444)

</lang>

Output:

The length of the sequence is 10003. The first 130 digits are:
0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350
and the last 130 digits are:
76898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000
Testing sequence: The sequence does contain all PINs.
Testing the reversed sequence: The sequence does contain all PINs.

After replacing 4444th digit with '.':
PIN 1459 does not occur in the sequence.
PIN 4591 does not occur in the sequence.
PIN 5814 does not occur in the sequence.
PIN 8145 does not occur in the sequence.
The sequence does not contain all PINs.

Kotlin

Translation of: Go

<lang scala>const val digits = "0123456789"

fun deBruijn(k: Int, n: Int): String {

   val alphabet = digits.substring(0, k)
   val a = ByteArray(k * n)
   val seq = mutableListOf<Byte>()
   fun db(t: Int, p: Int) {
       if (t > n) {
           if (n % p == 0) {
               seq.addAll(a.sliceArray(1..p).asList())
           }
       } else {
           a[t] = a[t - p]
           db(t + 1, p)
           var j = a[t - p] + 1
           while (j < k) {
               a[t] = j.toByte()
               db(t + 1, t)
               j++
           }
       }
   }
   db(1, 1)
   val buf = StringBuilder()
   for (i in seq) {
       buf.append(alphabet[i.toInt()])
   }
   val b = buf.toString()
   return b + b.subSequence(0, n - 1)

}

fun allDigits(s: String): Boolean {

   for (c in s) {
       if (c < '0' || '9' < c) {
           return false
       }
   }
   return true

}

fun validate(db: String) {

   val le = db.length
   val found = MutableList(10_000) { 0 }
   val errs = mutableListOf<String>()
   // Check all strings of 4 consecutive digits within 'db'
   // to see if all 10,000 combinations occur without duplication.
   for (i in 0 until le - 3) {
       val s = db.substring(i, i + 4)
       if (allDigits(s)) {
           val n = s.toInt()
           found[n]++
       }
   }
   for (i in 0 until 10_000) {
       if (found[i] == 0) {
           errs.add("    PIN number %04d missing".format(i))
       } else if (found[i] > 1) {
           errs.add("    PIN number %04d occurs %d times".format(i, found[i]))
       }
   }
   val lerr = errs.size
   if (lerr == 0) {
       println("  No errors found")
   } else {
       val pl = if (lerr == 1) {
           ""
       } else {
           "s"
       }
       println("  $lerr error$pl found:")
       println(errs.joinToString("\n"))
   }

}

fun main() {

   var db = deBruijn(10, 4)
   val le = db.length

   println("The length of the de Bruijn sequence is $le")
   println("\nThe first 130 digits of the de Bruijn sequence are: ${db.subSequence(0, 130)}")
   println("\nThe last 130 digits of the de Bruijn sequence are: ${db.subSequence(le - 130, le)}")

   println("\nValidating the deBruijn sequence:")
   validate(db)

   println("\nValidating the reversed deBruijn sequence:")
   validate(db.reversed())

   val bytes = db.toCharArray()
   bytes[4443] = '.'
   db = String(bytes)
   println("\nValidating the overlaid deBruijn sequence:")
   validate(db)

}</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the deBruijn sequence:
  No errors found

Validating the reversed deBruijn sequence:
  No errors found

Validating the overlaid deBruijn sequence:
  4 errors found:
    PIN number 1459 missing
    PIN number 4591 missing
    PIN number 5814 missing
    PIN number 8145 missing

Lua

Translation of: C++

<lang lua>function tprint(tbl)

   for i,v in pairs(tbl) do
       print(v)
   end

end

function deBruijn(k, n)

   local a = {}
   for i=1, k*n do
       table.insert(a, 0)
   end

   local seq = {}
   function db(t, p)
       if t > n then
           if n % p == 0 then
               for i=1, p do
                   table.insert(seq, a[i + 1])
               end
           end
       else
           a[t + 1] = a[t - p + 1]
           db(t + 1, p)

           local j = a[t - p + 1] + 1
           while j < k do
               a[t + 1] = j % 256
               db(t + 1, t)
               j = j + 1
           end
       end
   end

   db(1, 1)

   local buf = ""
   for i,v in pairs(seq) do
       buf = buf .. tostring(v)
   end
   return buf .. buf:sub(1, n - 1)

end

function allDigits(s)

   return s:match('[0-9]+') == s

end

function validate(db)

   local le = string.len(db)
   local found = {}
   local errs = {}

   for i=1, 10000 do
       table.insert(found, 0)
   end

   -- Check all strings of 4 consecutive digits within 'db'
   -- to see if all 10,000 combinations occur without duplication.
   for i=1, le - 3 do
       local s = db:sub(i, i + 3)
       if allDigits(s) then
           local n = tonumber(s)
           found[n + 1] = found[n + 1] + 1
       end
   end

   local count = 0
   for i=1, 10000 do
       if found[i] == 0 then
           table.insert(errs, "    PIN number " .. (i - 1) .. " missing")
           count = count + 1
       elseif found[i] > 1 then
           table.insert(errs, "    PIN number " .. (i - 1) .. " occurs " .. found[i] .. " times")
           count = count + 1
       end
   end

   if count == 0 then
       print("  No errors found")
   else
       tprint(errs)
   end

end

function main()

   local db = deBruijn(10,4)

   print("The length of the de Bruijn sequence is " .. string.len(db))
   print()

   io.write("The first 130 digits of the de Bruijn sequence are: ")
   print(db:sub(0, 130))
   print()

   io.write("The last 130 digits of the de Bruijn sequence are: ")
   print(db:sub(-130))
   print()

   print("Validating the de Bruijn sequence:")
   validate(db)
   print()

   print("Validating the reversed de Bruijn sequence:")
   validate(db:reverse())
   print()

   db = db:sub(1,4443) .. "." .. db:sub(4445)
   print("Validating the overlaid de Bruijn sequence:")
   validate(db)
   print()

end

main()</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the de Bruijn sequence:
  No errors found

Validating the reversed de Bruijn sequence:
  No errors found

Validating the overlaid de Bruijn sequence:
    PIN number 1459 missing
    PIN number 4591 missing
    PIN number 5814 missing
    PIN number 8145 missing

Mathematica/Wolfram Language

<lang Mathematica>seq = DeBruijnSequence[Range[0, 9], 4]; seq = seq~Join~Take[seq, 3]; Length[seq] {seq;; 130, seq-130 ;;} Complement[

StringDrop[ToString[NumberForm[#, 4, NumberPadding -> {"0", "0"}]], 
   1] & /@ Range[0, 9999], 
Union[StringJoin /@ Partition[ToString /@ seq, 4, 1]]]

seq = Reverse[seq]; Complement[

StringDrop[ToString[NumberForm[#, 4, NumberPadding -> {"0", "0"}]], 
   1] & /@ Range[0, 9999], 
Union[StringJoin /@ Partition[ToString /@ seq, 4, 1]]]

seq4444 = "."; Complement[

StringDrop[ToString[NumberForm[#, 4, NumberPadding -> {"0", "0"}]], 
   1] & /@ Range[0, 9999], 
Union[StringJoin /@ Partition[ToString /@ seq, 4, 1]]]</lang>

Output:

10003
{{0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 5, 0, 0,
   0, 6, 0, 0, 0, 7, 0, 0, 0, 8, 0, 0, 0, 9, 0, 0, 1, 1, 0, 0, 1, 2, 
  0, 0, 1, 3, 0, 0, 1, 4, 0, 0, 1, 5, 0, 0, 1, 6, 0, 0, 1, 7, 0, 0, 1,
   8, 0, 0, 1, 9, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 2, 3, 0, 0, 2, 4, 0, 
  0, 2, 5, 0, 0, 2, 6, 0, 0, 2, 7, 0, 0, 2, 8, 0, 0, 2, 9, 0, 0, 3, 1,
   0, 0, 3, 2, 0, 0, 3, 3, 0, 0, 3, 4, 0, 0, 3, 5, 0}, {6, 8, 9, 8, 6,
   8, 9, 9, 6, 9, 6, 9, 7, 7, 6, 9, 7, 8, 6, 9, 7, 9, 6, 9, 8, 7, 6, 
  9, 8, 8, 6, 9, 8, 9, 6, 9, 9, 7, 6, 9, 9, 8, 6, 9, 9, 9, 7, 7, 7, 7,
   8, 7, 7, 7, 9, 7, 7, 8, 8, 7, 7, 8, 9, 7, 7, 9, 8, 7, 7, 9, 9, 7, 
  8, 7, 8, 7, 9, 7, 8, 8, 8, 7, 8, 8, 9, 7, 8, 9, 8, 7, 8, 9, 9, 7, 9,
   7, 9, 8, 8, 7, 9, 8, 9, 7, 9, 9, 8, 7, 9, 9, 9, 8, 8, 8, 8, 9, 8, 
  8, 9, 9, 8, 9, 8, 9, 9, 9, 9, 0, 0, 0}}
{}
{}
{"1478", "4781", "7813", "8137"}

Nim

Translation of: D

<lang Nim>import algorithm, parseutils, strformat, strutils

const Digits = "0123456789"

---------------------------------------------------------------------------------------------------

func deBruijn(k, n: int): string =

 let alphabet = Digits[0..<k]
 var a = newSeq[byte](k * n)
 var sequence: seq[byte]

 #.................................................................................................

 func db(t, p: int) =
   if t > n:
     if n mod p == 0:
       sequence &= a[1..p]
   else:
     a[t] = a[t - p]
     db(t + 1, p)
     var j = a[t - p] + 1
     while j < k.uint:
       a[t] = j
       db(t + 1, t)
       inc j

   #...............................................................................................

 db(1, 1)
 for i in sequence:
   result &= alphabet[i]
 result &= result[0..(n-2)]

---------------------------------------------------------------------------------------------------

proc validate(db: string) =

 var found: array[10_000, int]
 var errs: seq[string]

 ## Check all strings of 4 consecutive digits within 'db'
 ## to see if all 10,000 combinations occur without duplication.
 for i in 0..(db.len - 4):
   let s = db[i..(i+3)]
   var n: int
   if s.parseInt(n) == 4:
     inc found[n]

 for n, count in found:
   if count == 0:
     errs &= fmt"    PIN number {n:04d} missing"
   elif count > 1:
     errs &= fmt"    PIN number {n:04d} occurs {count} times"

 if errs.len == 0:
   echo "  No errors found"
 else:
   let plural = if errs.len == 1: "" else: "s"
   echo fmt"  {errs.len} error{plural} found"
   for err in errs: echo err

———————————————————————————————————————————————————————————————————————————————————————————————————

var db = deBruijn(10, 4)

echo fmt"The length of the de Bruijn sequence is {db.len}" echo "" echo fmt"The first 130 digits of the de Bruijn sequence are: {db[0..129]}" echo "" echo fmt"The last 130 digits of the de Bruijn sequence are: {db[^130..^1]}" echo ""

echo "Validating the deBruijn sequence:" db.validate() echo "" echo "Validating the reversed deBruijn sequence:" reversed(db).join().validate() echo ""

db[4443] = '.' echo "Validating the overlaid deBruijn sequence:" db.validate()</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the deBruijn sequence:
  No errors found

Validating the reversed deBruijn sequence:
  No errors found

Validating the overlaid deBruijn sequence:
  4 errors found
    PIN number 1459 missing
    PIN number 4591 missing
    PIN number 5814 missing
    PIN number 8145 missing

Pascal

A console application in Free Pascal, created with the Lazarus IDE.

For a given word length n, constructs a de Bruijn sequence by concatenating, in lexicographic order, all the Lyndon words whose length divides n. (See Wikipedia article "de Bruijn sequence", section "Construction".) <lang pascal> program deBruijnSequence; uses SysUtils;

// Create a de Bruijn sequence for the given word length and alphabet. function deBruijn( const n : integer; // word length

                  const alphabet : string) : string;

var

 d, k, m, s, t, seqLen : integer;
 w : array of integer;

begin

 k := Length( alphabet);
 // de Bruijn sequence will have length k^n
 seqLen := 1;
 for t := 1 to n do seqLen := seqLen*k;
 SetLength( result, seqLen);
 d := 0; // index into de Bruijn sequence (will be pre-inc'd)
 // Work through Lyndon words of length <= n, in lexicographic order.
 SetLength( w, n); // w holds array of indices into the alphabet
 w[0] := 1; // first Lyndon word
 m := 1; // m = length of Lyndon word
 repeat
   // If m divides n, append the current Lyndon word to the output
   if (m = n) or (m = 1) or (n mod m = 0) then begin
     for t := 0 to m - 1 do begin
       inc(d);
       result[d] := alphabet[w[t]];
     end;
   end;
   // Get next Lyndon word using Duval's algorithm:
   // (1) Fill w with repetitions of current word
   s := 0; t := m;
   while (t < n) do begin
     w[t] := w[s];
     inc(t);  inc(s);
     if s = m then s := 0;
   end;
   // (2) Repeatedly delete highest index k from end of w, if present
   m := n;
   while (m > 0) and (w[m - 1] = k) do dec(m);
   // (3) If word is now null, stop; else increment end value
   if m > 0 then inc( w[m - 1]);
 until m = 0;
 Assert( d = seqLen); // check that the sequence is exactly filled in

end;

// Check a de Bruijn sequence, assuming that its alphabet consists // of the digits '0'..'9' (in any order); procedure CheckDecimal( const n : integer; // word length

                       const deB : string);

var

 count : array of integer;
 j, L, pin, nrErrors : integer;
 wrap : string;

begin

 L := Length( deB);
 // The de Bruijn sequence is cyclic; make an array to handle wrapround.
 SetLength( wrap, 2*n - 2);
 for j := 1 to n - 1 do wrap[j] := deB[L + j - n  + 1];
 for j := n to 2*n - 2 do wrap[j] := deB[j - n + 1];
 // Count occurrences of each PIN.
 // PIN = -1 if character is not a decimal digit.
 SetLength( count, L);
 for j := 0 to L - 1 do count[L] := 0;
 for j := 1 to L - n + 1 do begin
   pin := SysUtils.StrToIntDef( Copy( deB, j, n), -1);
   if pin >= 0 then inc( count[pin]);
 end;
 for j := 1 to n - 1 do begin
   pin := SysUtils.StrToIntDef( Copy( wrap, j, n), -1);
   if pin >= 0 then inc( count[pin]);
 end;
 // Check that all counts are 1
 nrErrors := 0;
 for j := 0 to L - 1 do begin
   if count[j] <> 1 then begin
     inc( nrErrors);
     WriteLn( SysUtils.Format( '  PIN %d has count %d', [j, count[j]]));
   end;
 end;
 WriteLn( SysUtils.Format( '  Number of errors = %d', [nrErrors]));
end;

// Main routine var

 deB, rev : string;
 L, j : integer;

begin

  deB := deBruijn( 4, '0123456789');

// deB := deBruijn( 4, '7368290514'); // any permutation would do

  L := Length( deB);
  WriteLn( SysUtils.Format( 'Length of de Bruijn sequence = %d', [L]));
  if L >= 260 then begin
    WriteLn;
    WriteLn( 'First and last 130 characters are:');
    WriteLn( Copy( deB, 1, 65));
    WriteLn( Copy( deb, 66, 65));
    WriteLn( '...');
    WriteLn( Copy( deB, L - 129, 65));
    WriteLn( Copy( deB, L - 64, 65));
  end;
  WriteLn;
  WriteLn( 'Checking de Bruijn sequence:');
  CheckDecimal( 4, deB);
  // Check reversed sequence
  SetLength( rev, L);
  for j := 1 to L do rev[j] := deB[L + 1 - j];
  WriteLn( 'Checking reversed sequence:');
  CheckDecimal( 4, rev);
  // Check sequence with '.' instad of decimal digit
  if L >= 4444 then begin
    deB[4444] := '.';
    WriteLn( 'Checking vandalized sequence:');
    CheckDecimal( 4, deB);
  end;

end. </lang>

Output:

Length of de Bruijn sequence = 10000

First and last 130 characters are:
00001000200030004000500060007000800090011001200130014001500160017
00180019002100220023002400250026002700280029003100320033003400350
...
89768986899696977697869796987698869896997699869997777877797788778
97798779978787978887889789878997979887989799879998888988998989999

Checking de Bruijn sequence:
  Number of errors = 0
Checking reversed sequence:
  Number of errors = 0
Checking vandalized sequence:
  PIN 1459 has count 0
  PIN 4591 has count 0
  PIN 5814 has count 0
  PIN 8145 has count 0
  Number of errors = 4

Perl

Translation of: Raku

<lang perl>use strict; use warnings; use feature 'say';

my $seq; for my $x (0..99) {

   my $a = sprintf '%02d', $x;
   next if substr($a,1,1) < substr($a,0,1);
   $seq .= (substr($a,0,1) == substr($a,1,1)) ? substr($a,0,1) : $a;
   for ($a+1 .. 99) {
       next if substr(sprintf('%02d', $_), 1,1) <= substr($a,0,1);
       $seq .= sprintf "%s%02d", $a, $_;
   }

} $seq .= '000';

sub check {

   my($seq) = @_;
   my %chk;
   for (0.. -1 + length $seq) { $chk{substr($seq, $_, 4)}++ }
   say 'Missing: ' . join ' ', grep { ! $chk{ sprintf('%04d',$_) } } 0..9999;
   say 'Extra:   ' . join ' ', sort grep { $chk{$_} > 1 } keys %chk;

}

my $n = 130; say "de Bruijn sequence length: " . length $seq; say "\nFirst $n characters:\n" . substr($seq, 0, $n ); say "\nLast $n characters:\n" . substr($seq, -$n, $n); say "\nIncorrect 4 digit PINs in this sequence:"; check $seq;

say "\nIncorrect 4 digit PINs in the reversed sequence:"; check(reverse $seq);

say "\nReplacing the 4444th digit, '@{[substr($seq,4443,1)]}', with '5'"; substr $seq, 4443, 1, 5; say "Incorrect 4 digit PINs in the revised sequence:"; check $seq;</lang>

Output:

de Bruijn sequence length: 10003

First 130 characters:
0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

Last 130 characters:
6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Incorrect 4 digit PINs in this sequence:
Missing:
Extra:

Incorrect 4 digit PINs in the reversed sequence:
Missing:
Extra:

Replacing the 4444th digit, '4', with '5'
Incorrect 4 digit PINs in the revised sequence:
Missing: 1459 4591 5814 8145
Extra:   1559 5591 5815 8155

Phix

Translation of: zkl

Translation of: Go

string deBruijn = ""
for n=0 to 99 do
    string a = sprintf("%02d",n)
    integer a1 = a[1],
            a2 = a[2]
    if a2>=a1 then
        deBruijn &= iff(a1=a2?a1:a)
        for m=n+1 to 99 do
            string ms = sprintf("%02d",m)
            if ms[2]>a1 then
                deBruijn &= a&ms
            end if
        end for
   end if
end for
deBruijn &= "000"
printf(1,"de Bruijn sequence length: %d\n\n",length(deBruijn))
printf(1,"First 130 characters:\n%s\n\n",deBruijn[1..130])
printf(1,"Last 130 characters:\n%s\n\n",deBruijn[-130..-1])
 
function check(string text)
    sequence res = {}
    sequence found = repeat(0,10000)
    integer k
    for i=1 to length(text)-3 do
        k = to_integer(text[i..i+3],-1)+1
        if k!=0 then found[k] += 1 end if
    end for
    for i=1 to 10000 do
        k = found[i]
        if k!=1 then
            string e = sprintf("Pin number %04d ",i-1)
            e &= iff(k=0?"missing":sprintf("occurs %d times",k))
            res = append(res,e)
        end if
    end for
    k = length(res)
    if k=0 then
        res = "No errors found"
    else
        string s = iff(k=1?"":"s")
        res = sprintf("%d error%s found:\n ",{k,s})&join(res,"\n ")
    end if
    return res
end function
 
printf(1,"Missing 4 digit PINs in this sequence: %s\n", check(deBruijn))
printf(1,"Missing 4 digit PINs in the reversed sequence: %s\n",check(reverse(deBruijn)))
printf(1,"4444th digit in the sequence: %c (setting it to .)\n", deBruijn[4444])
deBruijn[4444] = '.'
printf(1,"Re-running checks: %s\n",check(deBruijn))

Output:

de Bruijn sequence length: 10003

First 130 characters:
0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

Last 130 characters:
6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Missing 4 digit PINs in this sequence: No errors found
Missing 4 digit PINs in the reversed sequence: No errors found
4444th digit in the sequence: 4 (setting it to .)
Re-running checks: 4 errors found:
 Pin number 1459 missing
 Pin number 4591 missing
 Pin number 5814 missing
 Pin number 8145 missing

Python

from https://en.wikipedia.org/wiki/De_Bruijn_sequence

def de_bruijn(k, n):

   """
   de Bruijn sequence for alphabet k
   and subsequences of length n.
   """
   try:
       # let's see if k can be cast to an integer;
       # if so, make our alphabet a list
       _ = int(k)
       alphabet = list(map(str, range(k)))

   except (ValueError, TypeError):
       alphabet = k
       k = len(k)

   a = [0] * k * n
   sequence = []

   def db(t, p):
       if t > n:
           if n % p == 0:
               sequence.extend(a[1:p + 1])
       else:
           a[t] = a[t - p]
           db(t + 1, p)
           for j in range(a[t - p] + 1, k):
               a[t] = j
               db(t + 1, t)
   db(1, 1)
   return "".join(alphabet[i] for i in sequence)

def validate(db):

   """
   
   Check that all 10,000 combinations of 0-9 are present in 
   De Bruijn string db.
   
   Validating the reversed deBruijn sequence:
     No errors found
   
   Validating the overlaid deBruijn sequence:
     4 errors found:
       PIN number 1459 missing
       PIN number 4591 missing
       PIN number 5814 missing
       PIN number 8145 missing
   
   """
   
   dbwithwrap = db+db[0:3]
   
   digits = '0123456789'
   
   errorstrings = []
   
   for d1 in digits:
       for d2 in digits:
           for d3 in digits:
               for d4 in digits:
                   teststring = d1+d2+d3+d4
                   if teststring not in dbwithwrap:
                       errorstrings.append(teststring)
                       
   if len(errorstrings) > 0:
       print("  "+str(len(errorstrings))+" errors found:")
       for e in errorstrings:
           print("  PIN number "+e+"  missing")
   else:
       print("  No errors found")

db = de_bruijn(10, 4)

print(" ") print("The length of the de Bruijn sequence is ", str(len(db))) print(" ") print("The first 130 digits of the de Bruijn sequence are: "+db[0:130]) print(" ") print("The last 130 digits of the de Bruijn sequence are: "+db[-130:]) print(" ") print("Validating the deBruijn sequence:") validate(db) dbreversed = db[::-1] print(" ") print("Validating the reversed deBruijn sequence:") validate(dbreversed) dboverlaid = db[0:4443]+'.'+db[4444:] print(" ") print("Validating the overlaid deBruijn sequence:") validate(dboverlaid) </lang>

Output:

The length of the de Bruijn sequence is  10000
 
The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350
 
The last 130 digits of the de Bruijn sequence are: 8976898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999
 
Validating the deBruijn sequence:
  No errors found
 
Validating the reversed deBruijn sequence:
  No errors found
 
Validating the overlaid deBruijn sequence:
  4 errors found:
  PIN number 1459  missing
  PIN number 4591  missing
  PIN number 5814  missing
  PIN number 8145  missing

Racket

Translation of: Go

<lang racket>#lang racket

(define (de-bruijn k n)

 (define a (make-vector (* k n) 0))
 (define seq '())
 (define (db t p)
   (cond
     [(> t n) (when (= (modulo n p) 0)
                (set! seq (cons (call-with-values
                                 (thunk (vector->values a 1 (add1 p)))
                                 list)
                                seq)))]
     [else (vector-set! a t (vector-ref a (- t p)))
           (db (add1 t) p)
           (for ([j (in-range (add1 (vector-ref a (- t p))) k)])
             (vector-set! a t j)
             (db (add1 t) t))]))
 (db 1 1)
 (define seq* (append* (reverse seq)))
 (append seq* (take seq* (sub1 n))))

(define seq (de-bruijn 10 4)) (printf "The length of the de Bruijn sequence is ~a\n\n" (length seq)) (printf "The first 130 digits of the de Bruijn sequence are:\n~a\n\n"

       (take seq 130))

(printf "The last 130 digits of the de Bruijn sequence are:\n~a\n\n"

       (take-right seq 130))

(define (validate name seq)

 (printf "Validating the ~ade Bruijn sequence:\n" name)
 (define expected (for/set ([i (in-range 0 10000)]) i))
 (define actual (for/set ([a (in-list seq)]
                          [b (in-list (rest seq))]
                          [c (in-list (rest (rest seq)))]
                          [d (in-list (rest (rest (rest seq))))])
                  (+ (* 1000 a) (* 100 b) (* 10 c) d)))
 (define diff (set-subtract expected actual))
 (cond
   [(set-empty? diff) (printf "  No errors found\n")]
   [else (for ([n (in-set diff)])
           (printf "  ~a is missing\n" (~a n #:width 4 #:pad-string "0")))])
 (newline))

(validate "" seq) (validate "reversed " (reverse seq)) (validate "overlaid " (list-update seq 4443 add1))</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are:
(0 0 0 0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 8 0 0 0 9 0 0 1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 5 0 0 1 6 0 0 1 7 0 0 1 8 0 0 1 9 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0 2 6 0 0 2 7 0 0 2 8 0 0 2 9 0 0 3 1 0 0 3 2 0 0 3 3 0 0 3 4 0 0 3 5 0)

The last 130 digits of the de Bruijn sequence are:
(6 8 9 8 6 8 9 9 6 9 6 9 7 7 6 9 7 8 6 9 7 9 6 9 8 7 6 9 8 8 6 9 8 9 6 9 9 7 6 9 9 8 6 9 9 9 7 7 7 7 8 7 7 7 9 7 7 8 8 7 7 8 9 7 7 9 8 7 7 9 9 7 8 7 8 7 9 7 8 8 8 7 8 8 9 7 8 9 8 7 8 9 9 7 9 7 9 8 8 7 9 8 9 7 9 9 8 7 9 9 9 8 8 8 8 9 8 8 9 9 8 9 8 9 9 9 9 0 0 0)

Validating the de Bruijn sequence:
  No errors found

Validating the reversed de Bruijn sequence:
  No errors found

Validating the overlaid de Bruijn sequence:
  1459 is missing
  4591 is missing
  8145 is missing
  5814 is missing

Raku

(formerly Perl 6)

Works with: Rakudo version 2019.07.1

Deviates very slightly from the task spec. Generates a randomized de Bruijn sequence and replaces the 4444th digit with a the digit plus 1 mod 10 rather than a '.', mostly so it can demonstrate detection of extra PINs as well as missing ones.

<lang perl6># Generate the sequence my $seq;

for ^100 {

   my $a = .fmt: '%02d';
   next if $a.substr(1,1) < $a.substr(0,1);
   $seq ~= ($a.substr(0,1) == $a.substr(1,1)) ?? $a.substr(0,1) !! $a;
   for +$a ^..^ 100 {
       next if .fmt('%02d').substr(1,1) <= $a.substr(0,1);
       $seq ~= sprintf "%s%02d", $a, $_ ;
   }

}

$seq = $seq.comb.list.rotate((^10000).pick).join;

$seq ~= $seq.substr(0,3);

sub check ($seq) {

   my %chk;
   for ^($seq.chars) { %chk{$seq.substr( $_, 4 )}++ }
   put 'Missing: ', (^9999).grep( { not %chk{ .fmt: '%04d' } } ).fmt: '%04d';
   put 'Extra:   ', %chk.grep( *.value > 1 )».key.sort.fmt: '%04d';

}

1. The Task

put "de Bruijn sequence length: " ~ $seq.chars;

put "\nFirst 130 characters:\n" ~ $seq.substr( 0, 130 );

put "\nLast 130 characters:\n" ~ $seq.substr( * - 130 );

put "\nIncorrect 4 digit PINs in this sequence:"; check $seq;

put "\nIncorrect 4 digit PINs in the reversed sequence:"; check $seq.flip;

my $digit = $seq.substr(4443,1); put "\nReplacing the 4444th digit, ($digit) with { ($digit += 1) %= 10 }"; put "Incorrect 4 digit PINs in the revised sequence:"; $seq.substr-rw(4443,1) = $digit; check $seq;</lang>

Sample output:

de Bruijn sequence length: 10003

First 130 characters:
4558455945654566456745684569457545764577457845794585458645874588458945954596459745984599464647464846494655465646574658465946654666

Last 130 characters:
5445644574458445944654466446744684469447544764477447844794485448644874488448944954496449744984499454546454745484549455545564557455

Incorrect 4 digit PINs in this sequence:
Missing: 
Extra:   

Incorrect 4 digit PINs in the reversed sequence:
Missing: 
Extra:   

Replacing the 4444th digit, (1) with 2
Incorrect 4 digit PINs in the revised sequence:
Missing: 0961 1096 6109 9610
Extra:   0962 2096 6209 9620

REXX

The de Bruijn sequence generated by these REXX programs are identical to the sequence shown on the discussion page (1^st topic).

hard-coded node to be removed

<lang rexx>/*REXX pgm calculates the de Bruijn sequence for all pin numbers (4 digit decimals). */ $= /*initialize the de Bruijn sequence. */

=10; lastNode= (#-2)(#-2)(#-1)(#-2) /*this number is formed when this # ···*/

                                                /*  ··· is skipped near the cycle end. */
 do j=0  for 10;  $= $ || j;  jj= j || j        /*compose the left half of the numbers.*/
                                                /* [↓]     "  right  "   "  "     "    */
                               do k=jj+1  to 99;      z= jj || right(k, 2, 0)
                               if z==lastNode  then iterate    /*the last node skipped.*/
                               if pos(z, $)\==0  then iterate  /*# in sequence? Skip it*/
                               $= $ || z        /* ◄─────────────────────────────────┐ */
                               end   /*k*/      /*append a number to the sequence──◄─┘ */

    do r= jj  to (j || 9);  b= right(r, 2, 0)   /*compose the left half of the numbers.*/
    if b==jj  then iterate
    $= $ || right(b, 2, 0)                      /* [↓]     "  right  "   "  "     "    */
                               do k= b+1  to 99;      z= right(b, 2, 0) || right(k, 2, 0)
                               if pos(z, $)\==0  then iterate  /*# in sequence? Skip it*/
                               $= $ || z        /* ◄─────────────────────────────────┐ */
                               end   /*k*/      /*append a number to the sequence──◄─┘ */
    end   /*r*/
 end      /*j*/
                     @deB= 'de Bruijn sequence' /*literal used in some SAY instructions*/

$= $ || left($, 3) /*append 000*/ /*simulate "wrap-around" de Bruijn seq.*/

      say 'length of the' @deB " is " length($) /*display the length of  de Bruijn seq.*/

say; say 'first 130 digits of the' @deB":" /*display the title for the next line. */

      say left($, 130)                          /*display 130 left-most digits of seq. */

say; say ' last 130 digits of the' @deB":" /*display the title for the next line. */

      say right($, 130)                         /*display 130 right-most digits of seq.*/

say /*display a blank line. */ call val $ /*call the VAL sub for verification. */

              @deB= 'reversed'   @deB           /*next,  we'll check on a reversed seq.*/

$$= reverse($) /*do what a mirror does, reversify it.*/ call val $$ /*call the VAL sub for verification. */ $= overlay(., $, 4444) /*replace 4,444th digit with a period. */

              @deB= 'overlaid' subword(@deB, 2) /* [↑] this'll cause a validation barf.*/

call val $ /*call the VAL sub for verification. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ val: parse arg $$$; e= 0; _= copies('─',8) /*count of errors (missing PINs) so far*/

    say;      say _ 'validating the'    @deB"." /*display what's happening in the pgm. */
        do pin=0  for 1e4; pin4= right(pin,4,0) /* [↓]  maybe add leading zeros to pin.*/
        if pos(pin4, $$$)\==0  then iterate     /*Was number found?  Just as expected. */
        say 'PIN number '      pin       " wasn't found in"         @deb'.'
        e= e + 1                                /*bump the counter for number of errors*/
        end   /*pin*/                           /* [↑]  validate all 10,000 pin numbers*/
    if e==0  then e= 'No'                       /*Gooder English (sic) the error count.*/
    say _   e   'errors found.'                 /*display the number of errors found.  */
    return</lang>

output:

length of the de Bruijn sequence  is  10003

first 130 digits of the de Bruijn sequence:
0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

 last 130 digits of the de Bruijn sequence:
6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000


──────── validating the de Bruijn sequence.
──────── No errors found.

──────── validating the reversed de Bruijn sequence.
──────── No errors found.

──────── validating the overlaid de Bruijn sequence.
PIN number  1459  wasn't found in overlaid de Bruijn sequence.
PIN number  4591  wasn't found in overlaid de Bruijn sequence.
PIN number  5814  wasn't found in overlaid de Bruijn sequence.
PIN number  8145  wasn't found in overlaid de Bruijn sequence.
──────── 4 errors found.

programmatically removing of a node

Programming note: instead of hardcoding the lastNode (that is elided from the sequence), the 5^th to the last node could simply be deleted.

This method slightly bloats the program and slows execution. <lang rexx>/*REXX pgm calculates the de Bruijn sequence for all pin numbers (4 digit decimals). */ $= /*initialize the de Bruijn sequence. */

  do j=0  for 10;  $= $ j;   jj= j || j          /*compose the left half of the numbers.*/
 $$= space($, 0)                                /* [↓]     "  right  "   "  "     "    */
                               do k=jj+1  to 99;      z= jj || right(k, 2, 0)
                               if pos(z, $$)\==0  then iterate /*# in sequence? Skip it*/
                               $= $ z           /* ◄─────────────────────────────────┐ */
                               end   /*k*/      /*append a number to the sequence──◄─┘ */
 $$= space($, 0)
    do r= jj  to (j || 9);  b= right(r, 2, 0)   /*compose the left half of the numbers.*/
    if b==jj  then iterate
    $= $ right(b, 2, 0)                         /* [↓]     "  right  "   "  "     "    */
    $$= space($, 0);           do k= b+1  to 99;      z= right(b, 2, 0) || right(k, 2, 0)
                               if pos(z, $$)\==0  then iterate /*# in sequence? Skip it*/
                               $= $ z           /* ◄─────────────────────────────────┐ */
                               end   /*k*/      /*append a number to the sequence──◄─┘ */
    $$= space($, 0)
    end   /*r*/
 end      /*j*/

$= delword($, words($)-4, 1) /*delete 5th from the last word in $. */ $= space($, 0)

                     @deB= 'de Bruijn sequence' /*literal used in some SAY instructions*/

$= $ || left($, 3) /*append 000*/ /*simulate "wrap-around" de Bruijn seq.*/

      say 'length of the' @deB " is " length($) /*display the length of  de Bruijn seq.*/

say; say 'first 130 digits of the' @deB":" /*display the title for the next line. */

      say left($, 130)                          /*display 130 left-most digits of seq. */

say; say ' last 130 digits of the' @deB":" /*display the title for the next line. */

      say right($, 130)                         /*display 130 right-most digits of seq.*/

call val $ /*call the VAL sub for verification. */

              @deB= 'reversed'   @deB           /*next,  we'll check on a reversed seq.*/

$r= reverse($) /*do what a mirror does, reversify it.*/ call val $r /*call the VAL sub for verification. */ $= overlay(., $, 4444) /*replace 4,444th digit with a period. */

              @deB= 'overlaid' subword(@deB, 2) /* [↑] this'll cause a validation barf.*/

call val $ /*call the VAL sub for verification. */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ val: parse arg $$$; e= 0; _= copies('─',8) /*count of errors (missing PINs) so far*/

    say;      say _ 'validating the'    @deB"." /*display what's happening in the pgm. */
        do pin=0  for 1e4; pin4= right(pin,4,0) /* [↓]  maybe add leading zeros to pin.*/
        if pos(pin4, $$$)\==0  then iterate     /*Was number found?  Just as expected. */
        say 'PIN number '      pin       " wasn't found in"         @deb'.'
        e= e + 1                                /*bump the counter for number of errors*/
        end   /*pin*/                           /* [↑]  validate all 10,000 pin numbers*/
    if e==0  then e= 'No'                       /*Gooder English (sic) the error count.*/
    say _   e   'errors found.'                 /*display the number of errors found.  */
    return</lang>

output is identical to the 1^st REXX version.

Ruby

Translation of: D

<lang ruby>def deBruijn(k, n)

   alphabet = "0123456789"
   @a = Array.new(k * n, 0)
   @seq = []

   def db(k, n, t, p)
       if t > n then
           if n % p == 0 then
               temp = @a[1 .. p]
               @seq.concat temp
           end
       else
           @a[t] = @a[t - p]
           db(k, n, t + 1, p)
           j = @a[t - p] + 1
           while j < k do
               @a[t] = j # & 0xFF
               db(k, n, t + 1, t)
               j = j + 1
           end
       end
   end
   db(k, n, 1, 1)

   buf = ""
   for i in @seq
       buf <<= alphabet[i]
   end
   return buf + buf[0 .. n-2]

end

def validate(db)

   le = db.length
   found = Array.new(10000, 0)
   errs = []
   # Check all strings of 4 consecutive digits within 'db'
   # to see if all 10,000 combinations occur without duplication.
   for i in 0 .. le-4
       s = db[i .. i+3]
       if s.scan(/\D/).empty? then
           found[s.to_i] += 1
       end
   end
   for i in 0 .. found.length - 1
       if found[i] == 0 then
           errs <<= ("    PIN number %04d missing" % [i])
       elsif found[i] > 1 then
           errs <<= ("    PIN number %04d occurs %d times" % [i, found[i]])
       end
   end
   if errs.length == 0 then
       print "  No errors found\n"
   else
       pl = (errs.length == 1) ? "" : "s"
       print "  ", errs.length, " error", pl, " found:\n"
       for err in errs
           print err, "\n"
       end
   end

end

db = deBruijn(10, 4) print "The length of the de Bruijn sequence is ", db.length, "\n\n" print "The first 130 digits of the de Bruijn sequence are: ", db[0 .. 129], "\n\n" print "The last 130 digits of the de Bruijn sequence are: ", db[-130 .. db.length], "\n\n"

print "Validating the de Bruijn sequence:\n" validate(db) print "\n"

db[4443] = '.' print "Validating the overlaid de Bruijn sequence:\n" validate(db)</lang>

Output:

The length of the de Bruijn sequence is 10003

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the de Bruijn sequence:
  No errors found

Validating the overlaid de Bruijn sequence:
  4 errors found:
    PIN number 1459 missing
    PIN number 4591 missing
    PIN number 5814 missing
    PIN number 8145 missing

Visual Basic .NET

Translation of: C#

<lang vbnet>Imports System.Text

Module Module1

   ReadOnly DIGITS As String = "0123456789"

   Function DeBruijn(k As Integer, n As Integer) As String
       Dim alphabet = DIGITS.Substring(0, k)
       Dim a(k * n) As Byte
       Dim seq As New List(Of Byte)
       Dim db As Action(Of Integer, Integer) = Sub(t As Integer, p As Integer)
                                                   If t > n Then
                                                       If n Mod p = 0 Then
                                                           Dim seg = New ArraySegment(Of Byte)(a, 1, p)
                                                           seq.AddRange(seg)
                                                       End If
                                                   Else
                                                       a(t) = a(t - p)
                                                       db(t + 1, p)
                                                       Dim j = a(t - p) + 1
                                                       While j < k
                                                           a(t) = j
                                                           db(t + 1, t)
                                                           j += 1
                                                       End While
                                                   End If
                                               End Sub
       db(1, 1)
       Dim buf As New StringBuilder
       For Each i In seq
           buf.Append(alphabet(i))
       Next
       Dim b = buf.ToString
       Return b + b.Substring(0, n - 1)
   End Function

   Function AllDigits(s As String) As Boolean
       For Each c In s
           If c < "0" OrElse "9" < c Then
               Return False
           End If
       Next
       Return True
   End Function

   Sub Validate(db As String)
       Dim le = db.Length
       Dim found(10000) As Integer
       Dim errs As New List(Of String)
       ' Check all strings of 4 consecutive digits within 'db'
       ' to see if all 10,000 combinations occur without duplication.
       For i = 1 To le - 3
           Dim s = db.Substring(i - 1, 4)
           If (AllDigits(s)) Then
               Dim n As Integer = Nothing
               Integer.TryParse(s, n)
               found(n) += 1
           End If
       Next
       For i = 1 To 10000
           If found(i - 1) = 0 Then
               errs.Add(String.Format("    PIN number {0,4} missing", i - 1))
           ElseIf found(i - 1) > 1 Then
               errs.Add(String.Format("    PIN number {0,4} occurs {1} times", i - 1, found(i - 1)))
           End If
       Next
       Dim lerr = errs.Count
       If lerr = 0 Then
           Console.WriteLine("  No errors found")
       Else
           Dim pl = If(lerr = 1, "", "s")
           Console.WriteLine("  {0} error{1} found:", lerr, pl)
           errs.ForEach(Sub(x) Console.WriteLine(x))
       End If
   End Sub

   Function Reverse(s As String) As String
       Dim arr = s.ToCharArray
       Array.Reverse(arr)
       Return New String(arr)
   End Function

   Sub Main()
       Dim db = DeBruijn(10, 4)
       Dim le = db.Length

       Console.WriteLine("The length of the de Bruijn sequence is {0}", le)
       Console.WriteLine(vbNewLine + "The first 130 digits of the de Bruijn sequence are: {0}", db.Substring(0, 130))
       Console.WriteLine(vbNewLine + "The last 130 digits of the de Bruijn sequence are: {0}", db.Substring(le - 130, 130))

       Console.WriteLine(vbNewLine + "Validating the deBruijn sequence:")
       Validate(db)

       Console.WriteLine(vbNewLine + "Validating the reversed deBruijn sequence:")
       Validate(Reverse(db))

       Dim bytes = db.ToCharArray
       bytes(4443) = "."
       db = New String(bytes)
       Console.WriteLine(vbNewLine + "Validating the overlaid deBruijn sequence:")
       Validate(db)
   End Sub

End Module</lang>

Output:

The first 130 digits of the de Bruijn sequence are: 0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

The last 130 digits of the de Bruijn sequence are: 6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Validating the deBruijn sequence:
  No errors found

Validating the reversed deBruijn sequence:
  No errors found

Validating the overlaid deBruijn sequence:
  4 errors found:
    PIN number 1459 missing
    PIN number 4591 missing
    PIN number 5814 missing
    PIN number 8145 missing

Wren

Translation of: Phix

Library: Wren-fmt

Library: Wren-str

<lang ecmascript>import "/fmt" for Fmt import "/str" for Str

var deBruijn = "" for (n in 0..99) {

   var a = Fmt.rjust(2, n, "0")
   var a1 = a[0].bytes[0]
   var a2 = a[1].bytes[0]
   if (a2 >= a1) {
       deBruijn = deBruijn + ((a1 == a2) ? String.fromByte(a1): a)
       var m = n + 1
       while (m <= 99) {
           var ms = Fmt.rjust(2, m, "0")
           if (ms[1].bytes[0] > a1) deBruijn = deBruijn + a + ms
           m = m + 1
       }
   }

}

deBruijn = deBruijn + "000" System.print("de Bruijn sequence length: %(deBruijn.count)\n") System.print("First 130 characters:\n%(deBruijn[0...130])\n") System.print("Last 130 characters:\n%(deBruijn[-130..-1])\n")

var check = Fn.new { |text|

   var res = []
   var found = List.filled(10000, 0)
   var k = 0
   for (i in 0...(text.count-3)) {
       var s = text[i..i+3]
       if (Str.allDigits(s)) {
           k = Num.fromString(s)
           found[k] = found[k] + 1
       }
   }
   for (i in 0...10000) {
       k = found[i]
       if (k != 1) {
           var e = "  Pin number %(Fmt.dz(4, i)) "
           e = e + ((k == 0) ? "missing" : "occurs %(k) times")
           res.add(e)
       }
   }
   k = res.count
   if (k == 0) {
       res = "No errors found"
   } else {
       var s = (k == 1) ? "" : "s"
       res = "%(k) error%(s) found:\n" + res.join("\n")
   }
   return res

}

System.print("Missing 4 digit PINs in this sequence: %(check.call(deBruijn))") System.print("Missing 4 digit PINs in the reversed sequence: %(check.call(deBruijn[-1..0]))")

System.print("\n4,444th digit in the sequence: '%(deBruijn[4443])' (setting it to '.')") deBruijn = deBruijn[0..4442] + "." + deBruijn[4444..-1] System.print("Re-running checks: %(check.call(deBruijn))")</lang>

Output:

de Bruijn sequence length: 10003

First 130 characters:
0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

Last 130 characters:
6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Missing 4 digit PINs in this sequence: No errors found
Missing 4 digit PINs in the reversed sequence: No errors found

4,444th digit in the sequence: '4' (setting it to '.')
Re-running checks: 4 errors found:
  Pin number 1459 missing
  Pin number 4591 missing
  Pin number 5814 missing
  Pin number 8145 missing

zkl

Translation of: Raku

<lang zkl>dbSeq:=Data(); // a byte/character buffer foreach n in (100){

  a,a01,a11 := "%02d".fmt(n), a[0,1], a[1,1];
  if(a11<a01) continue;
  dbSeq.append( if(a01==a11) a01 else a );
  foreach m in ([n+1 .. 99]){
     if("%02d".fmt(m)[1,1] <= a01) continue;
     dbSeq.append("%s%02d".fmt(a,m));
  }

} dbSeq.append("000");</lang> <lang zkl>seqText:=dbSeq.text; println("de Bruijn sequence length: ",dbSeq.len());

println("\nFirst 130 characters:\n",seqText[0,130]); println("\nLast 130 characters:\n", seqText[-130,*]);

fcn chk(seqText){

  chk:=Dictionary();
  foreach n in ([0..seqText.len()-1]){ chk[seqText[n,4]]=True }
  (9999).pump(List,"%04d".fmt,'wrap(k){ if(chk.holds(k)) Void.Skip else k })

} println("\nMissing 4 digit PINs in this sequence: ", chk(seqText).concat(" ")); print("Missing 4 digit PINs in the reversed sequence: ",chk(seqText.reverse()).concat(" "));

println("\n4444th digit in the sequence: ", seqText[4443]); dbSeq[4443]="."; println("Setting the 4444th digit and reruning checks: ",chk(dbSeq.text).concat(" "));</lang>

Output:

de Bruijn sequence length: 10003

First 130 characters:
0000100020003000400050006000700080009001100120013001400150016001700180019002100220023002400250026002700280029003100320033003400350

Last 130 characters:
6898689969697769786979698769886989699769986999777787779778877897798779978787978887889789878997979887989799879998888988998989999000

Missing 4 digit PINs in this sequence: 
Missing 4 digit PINs in the reversed sequence: 
4444th digit in the sequence: 4
Setting the 4444th digit and reruning checks: 1459 4591 5814 8145