Riordan numbers
Riordan numbers show up in several places in set theory. They are closely related to Motzkin numbers, and may be used to derive them.
Riordan numbers comprise the sequence a where:
a(0) = 1, a(1) = 0, for subsequent terms, a(n) = (n-1)*(2*a(n-1) + 3*a(n-2))/(n+1)
There are other generating functions, and you are free to use one most convenient for your language.
- Task
- Find and display the first 32 Riordan numbers.
- Stretch
- Find and display the digit count of the 1,000th Riordan number.
- Find and display the digit count of the 10,000th Riordan number.
- See also
ALGOL 68
Uses ALGOL 68G's LONG LONG INT which has programmer-specifiable precision. ALGOL 68G version 3 issues a warning that precision 5000 will impact performance but it still executes this program somewhat faster than version 2 does. <lang algol68>BEGIN # find some Riordan numbers #
# returns a table of the Riordan numbers 0 .. n #
OP RIORDAN = ( INT n )[]LONG INT:
BEGIN
[ 0 : n ]LONG INT a;
IF n >= 0 THEN
a[ 0 ] := 1;
IF n >= 1 THEN
a[ 1 ] := 0;
FOR i FROM 2 TO UPB a DO
a[ i ] := ( ( i - 1 )
* ( ( 2 * a[ i - 1 ] )
+ ( 3 * a[ i - 2 ] )
)
)
OVER ( i + 1 )
OD
FI
FI;
a
END # RIORDAN # ;
# returns a string representation of n with commas #
PROC commatise = ( STRING unformatted )STRING:
BEGIN
STRING result := "";
INT ch count := 0;
FOR c FROM UPB unformatted BY -1 TO LWB unformatted DO
IF ch count <= 2 THEN
ch count +:= 1
ELSE
ch count := 1;
IF unformatted[ c ] = " " THEN " " ELSE "," FI +=: result
FI;
unformatted[ c ] +=: result
OD;
result
END; # commatise #
# returns the length of s #
OP LENGTH = ( STRING s )INT: ( UPB s - LWB s ) + 1;
BEGIN # show some Riordann numbers #
[]LONG INT r = RIORDAN 31;
INT shown := 0;
FOR i FROM LWB r TO UPB r DO
print( ( commatise( whole( r[ i ], -15 ) ) ) );
IF ( shown +:= 1 ) = 4 THEN
print( ( newline ) );
shown := 0
FI
OD
END;
BEGIN # calculate the length of the 1 000th and 10 000th Riordan numbers #
PR precision 5000 PR # allow up to 5 000 digits for LONG LONG INT #
LONG LONG INT r2 := -1, r1 := 1, r := 0;
print( ( newline ) );
FOR i FROM 2 TO 9 999 DO
r2 := r1;
r1 := r;
r := ( ( i - 1 )
* ( ( 2 * r1 )
+ ( 3 * r2 )
)
)
OVER ( i + 1 );
IF i = 999 OR i = 9 999 THEN
STRING rs = whole( r, 0 )[ @ 1 ];
print( ( "The ", whole( i + 1, -6 ), "th number is: "
, rs[ 1 : 20 ], "...", rs[ LENGTH rs - 19 : ]
, " with ", whole( LENGTH rs, -5 ), " digits"
, newline
)
)
FI
OD
END
END</lang>
- Output:
1 0 1 1
3 6 15 36
91 232 603 1,585
4,213 11,298 30,537 83,097
227,475 625,992 1,730,787 4,805,595
13,393,689 37,458,330 105,089,229 295,673,994
834,086,421 2,358,641,376 6,684,761,125 18,985,057,351
54,022,715,451 154,000,562,758 439,742,222,071 1,257,643,249,140
The 1000th number is: 51077756867821111314...79942013897484633052 with 472 digits
The 10000th number is: 19927418577260688844...71395322020211157137 with 4765 digits
F#
<lang fsharp> // Riordan numbers. Nigel Galloway: August 19th., 2022 let r()=seq{yield 1I; yield 0I; yield! Seq.unfold(fun(n,n1,n2)->let r=(n-1I)*(2I*n1+3I*n2)/(n+1I) in Some(r,(n+1I,r,n1)))(2I,0I,1I)} let n=r()|>Seq.take 10000|>Array.ofSeq in n|>Array.take 32|>Seq.iter(printf "%A "); printfn "\nr[999] has %d digits\nr[9999] has %d digits" ((string n.[999]).Length) ((string n.[9999]).Length) </lang>
- Output:
1 0 1 1 3 6 15 36 91 232 603 1585 4213 11298 30537 83097 227475 625992 1730787 4805595 13393689 37458330 105089229 95673994 834086421 2358641376 6684761125 18985057351 54022715451 154000562758 439742222071 1257643249140 r[999] has 472 digits r[9999] has 4765 digits
J
Sequence extender:<lang J>riordanext=: (, (<: % >:)@# * 3 2 +/ .* _2&{.)</lang> Task example:<lang J> riordanext^:(30) 1x 0 1 0 1 1 3 6 15 36 91 232 603 1585 4213 11298 30537 83097 227475 625992 1730787 4805595 13393689 37458330 105089229 295673994 834086421 2358641376 6684761125 18985057351 54022715451 154000562758 439742222071 1257643249140</lang>Stretch:<lang J> #":(1e3-1){riordanext^:(1e3) x:1 0 472
#":(1e4-1){riordanext^:(1e4) x:1 0
4765</lang>
Phix
with javascript_semantics requires("1.0.2") -- mpz_get_str(comma_fill) was not working [!!!] include mpfr.e constant limit = 10000 sequence a = {mpz_init(1),mpz_init(0)} for n=2 to limit do mpz an = mpz_init() mpz_mul_si(an,a[n],2) mpz_addmul_si(an,a[n-1],3) mpz_mul_si(an,an,n-1) assert(mpz_fdiv_q_ui(an,an,n+1)=0) a &= an end for printf(1,"First 32 Riordan numbers:\n%s\n", {join_by(apply(true,mpz_get_str,{a[1..32],10,true}),1,4," ",fmt:="%17s")}) for i in {1e3, 1e4} do printf(1,"The %6s: %s\n", {ordinal(i), mpz_get_short_str(a[i])}) end for
- Output:
First 32 Riordan numbers:
1 0 1 1
3 6 15 36
91 232 603 1,585
4,213 11,298 30,537 83,097
227,475 625,992 1,730,787 4,805,595
13,393,689 37,458,330 105,089,229 295,673,994
834,086,421 2,358,641,376 6,684,761,125 18,985,057,351
54,022,715,451 154,000,562,758 439,742,222,071 1,257,643,249,140
The one thousandth: 51077756867821111314...79942013897484633052 (472 digits)
The ten thousandth: 19927418577260688844...71395322020211157137 (4,765 digits)
Raku
<lang perl6>use Lingua::EN::Numbers;
my @riordan = 1, 0, { state $n = 1; (++$n - 1) / ($n + 1) × (3 × $^a + 2 × $^b) } … *;
my $upto = 32; say "First {$upto.&cardinal} Riordan numbers:\n" ~ @riordan[^$upto]».&comma».fmt("%17s").batch(4).join("\n") ~ "\n";
sub abr ($_) { .chars < 41 ?? $_ !! .substr(0,20) ~ '..' ~ .substr(*-20) ~ " ({.chars} digits)" }
say "The {.Int.&ordinal}: " ~ abr @riordan[$_ - 1] for 1e3, 1e4</lang>
- Output:
First thirty-two Riordan numbers:
1 0 1 1
3 6 15 36
91 232 603 1,585
4,213 11,298 30,537 83,097
227,475 625,992 1,730,787 4,805,595
13,393,689 37,458,330 105,089,229 295,673,994
834,086,421 2,358,641,376 6,684,761,125 18,985,057,351
54,022,715,451 154,000,562,758 439,742,222,071 1,257,643,249,140
The one thousandth: 51077756867821111314..79942013897484633052 (472 digits)
The ten thousandth: 19927418577260688844..71395322020211157137 (4765 digits)
Wren
<lang ecmascript>import "./gmp" for Mpz import "./fmt" for Fmt
var limit = 10000 var a = List.filled(limit, null) a[0] = Mpz.one a[1] = Mpz.zero for (n in 2...limit) {
a[n] = (a[n-1] * 2 + a[n-2] * 3) * (n-1) / (n+1)
} System.print("First 32 Riordan numbers:") Fmt.tprint("$,17i", a[0..31], 4) System.print() for (i in [1e3, 1e4]) {
Fmt.print("$,8r: $20a ($,d digits)", i, a[i-1], a[i-1].toString.count)
}</lang>
- Output:
First 32 Riordan numbers:
1 0 1 1
3 6 15 36
91 232 603 1,585
4,213 11,298 30,537 83,097
227,475 625,992 1,730,787 4,805,595
13,393,689 37,458,330 105,089,229 295,673,994
834,086,421 2,358,641,376 6,684,761,125 18,985,057,351
54,022,715,451 154,000,562,758 439,742,222,071 1,257,643,249,140
1,000th: 51077756867821111314...79942013897484633052 (472 digits)
10,000th: 19927418577260688844...71395322020211157137 (4,765 digits)