Jacobsthal numbers

Jacobsthal numbers are an integer sequence related to Fibonacci numbers. Similar to Fibonacci, where each term is the sum of the previous two terms, each term is the sum of the previous, plus twice the one before that. Traditionally the sequence starts with the given terms 0, 1.


   J₀ = 0
   J₁ = 1
   J_n = J_n-1 + 2 × J_n-2

Terms may be calculated directly using one of several possible formulas:


   J_n = ( 2ⁿ - (-1)ⁿ ) / 3

Jacobsthal-Lucas numbers are very similar. They have the same recurrence relationship, the only difference is an initial starting value J₀ = 2 rather than J₀ = 0.

Terms may be calculated directly using one of several possible formulas:

   
   JL_n = 2ⁿ + (-1)ⁿ

Jacobsthal oblong numbers is the sequence obtained from multiplying each Jacobsthal number J_n by its direct successor J_n+1.

Jacobsthal primes are Jacobsthal numbers that are prime.

Task

Find and display the first 30 Jacobsthal numbers
Find and display the first 30 Jacobsthal-Lucas numbers
Find and display the first 20 Jacobsthal oblong numbers
Find and display at least the first 10 Jacobsthal primes

See also

Wikipedia: Jacobsthal number Numbers Aplenty - Jacobsthal number OEIS:A001045 - Jacobsthal sequence (or Jacobsthal numbers) OEIS:A014551 - Jacobsthal-Lucas numbers. OEIS:A084175 - Jacobsthal oblong numbers OEIS:A049883 - Primes in the Jacobsthal sequence Related task: Fibonacci sequence Related task: Leonardo numbers

ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Library: ALGOL 68-primes

<lang algol68>BEGIN # find some Jacobsthal and related Numbers #

   INT max jacobsthal = 29;        # highest Jacobsthal number we will find #
   INT max oblong     = 20; # highest Jacobsthal oblong number we will find #
   INT max j prime    = 20;     # number of Jacobsthal prinmes we will find #
   PR precision 200 PR                 # set the precision of LONG LONG INT #
   PR read "primes.incl.a68" PR                   # include prime utilities #
   [ 0 : max jacobsthal ]LONG INT j;         # will hold Jacobsthal numbers #
   [ 0 : max jacobsthal ]LONG INT jl;  # will hold Jacobsthal-Lucas numbers #
   [ 1 : max oblong     ]LONG INT jo; # will hold Jacobsthal oblong numbers #
   # calculate the Jacobsthal Numbers and related numbers                   #
   # Jacobsthal      : J0  = 0, J1  = 1, Jn  = Jn-1  + 2 × Jn-2             #
   # Jacobsthal-Lucas: JL0 = 2, JL1 = 1, JLn = JLn-1 + 2 × JLn-2            #
   # Jacobsthal oblong: JOn = Jn x Jn-1                                     #
   j[ 0 ] := 0; j[ 1 ] := 1; jl[ 0 ] := 2; jl[ 1 ] := 1; jo[ 1 ] := 0;
   FOR n FROM 2 TO UPB j DO
       j[  n ] := j[  n - 1 ] + ( 2 * j[  n - 2 ] );
       jl[ n ] := jl[ n - 1 ] + ( 2 * jl[ n - 2 ] )
   OD;
   FOR n TO UPB jo DO
       jo[ n ] := j[ n ] * j[ n - 1 ]
   OD;
   # prints an array of numbers with the specified legend                   #
   PROC show numbers = ( STRING legend, []LONG INT numbers )VOID:
        BEGIN
           INT n count := 0;
           print( ( "First ", whole( ( UPB numbers - LWB numbers ) + 1, 0 ), " ", legend, newline ) );
           FOR n FROM LWB numbers TO UPB numbers DO
               print( ( " ", whole( numbers[ n ], -11 ) ) );
               IF ( n count +:= 1 ) MOD 5 = 0 THEN print( ( newline ) ) FI
           OD
        END # show numbers # ;
   # show the various numbers numbers                                       #
   show numbers( "Jacobsthal Numbers:",        j  );
   show numbers( "Jacobsthal-Lucas Numbers:",  jl );
   show numbers( "Jacobsthal oblong Numbers:", jo );
   # find some prime Jacobsthal numbers                                     #
   LONG LONG INT  jn1 := j[ 1 ], jn2 := j[ 0 ];
   INT  p count := 0;
   print( ( "First ", whole( max j prime, 0 ), " Jacobstal primes:", newline ) );
   print( ( "   n  Jn", newline ) );
   FOR n FROM 2 WHILE p count < max j prime DO
       LONG LONG INT jn = jn1 + ( 2 * jn2 );
       jn2        := jn1;
       jn1        := jn;
       IF is probably prime( jn ) THEN
           # have a probably prime Jacobsthal number                        #
           p count +:= 1;
           print( ( whole( n, -4 ), ": ", whole( jn, 0 ), newline ) )
       FI
   OD

END</lang>

Output:

First 30 Jacobsthal Numbers:
           0           1           1           3           5
          11          21          43          85         171
         341         683        1365        2731        5461
       10923       21845       43691       87381      174763
      349525      699051     1398101     2796203     5592405
    11184811    22369621    44739243    89478485   178956971
First 30 Jacobsthal-Lucas Numbers:
           2           1           5           7          17
          31          65         127         257         511
        1025        2047        4097        8191       16385
       32767       65537      131071      262145      524287
     1048577     2097151     4194305     8388607    16777217
    33554431    67108865   134217727   268435457   536870911
First 20 Jacobsthal oblong Numbers:
           0           1           3          15          55
         231         903        3655       14535       58311
      232903      932295     3727815    14913991    59650503
   238612935   954429895  3817763271 15270965703 61084037575
First 20 Jacobstal primes:
   n  Jn
   3: 3
   4: 5
   5: 11
   7: 43
  11: 683
  13: 2731
  17: 43691
  19: 174763
  23: 2796203
  31: 715827883
  43: 2932031007403
  61: 768614336404564651
  79: 201487636602438195784363
 101: 845100400152152934331135470251
 127: 56713727820156410577229101238628035243
 167: 62357403192785191176690552862561408838653121833643
 191: 1046183622564446793972631570534611069350392574077339085483
 199: 267823007376498379256993682056860433753700498963798805883563
 313: 5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
 347: 95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443

AppleScript

Procedural

<lang applescript>on jacobsthalNumbers(variant, n)

   -- variant: text containing "Lucas", "oblong", or "prime" — or none of these.
   -- n: length of output sequence required.
   
   -- The two Jacobsthal numbers preceding the current 'j'. Initially the first two in the sequence.
   set {anteprev, prev} to {0, 1}
   -- Default plug-in script. Its handler simply appends the current 'j' to the output.
   script o
       property output : {anteprev, prev}
       on append(dummy, j)
           set end of output to j
       end append
   end script
   
   -- If a variant sequence is specified, change the first value or substitute
   -- a script whose handler decides the values to append to the output. 
   ignoring case
       if (variant contains "Lucas") then
           set anteprev to 2
           set o's output's first item to anteprev
       else if (variant contains "oblong") then
           script
               property output : {0}
               on append(prev, j)
                   set end of output to prev * j
               end append
           end script
           set o to result
       else if (variant contains "prime") then
           script
               property output : {}
               on append(dummy, j)
                   if (isPrime(j)) then set end of output to j
               end append
           end script
           set o to result
       end if
   end ignoring
   
   -- Work through the Jacobsthal process until the required output length is obtained.
   repeat until ((count o's output) = n)
       set j to anteprev + anteprev + prev
       tell o to append(prev, j)
       set anteprev to prev
       set prev to j
   end repeat
   
   return o's output

end jacobsthalNumbers

on isPrime(n)

   if (n < 3) then return (n is 2)
   if (n mod 2 is 0) then return false
   repeat with i from 3 to (n ^ 0.5) div 1 by 2
       if (n mod i is 0) then return false
   end repeat
   return true

end isPrime

-- Task and presentation of results!: on intToText(n)

   set txt to ""
   repeat until (n < 100000000)
       set txt to text 2 thru 9 of (100000000 + (n mod 100000000) div 1 as text) & txt
       set n to n div 100000000
   end repeat
   return (n as integer as text) & txt

end intToText

on chopList(theList, sublistLen)

   script o
       property lst : theList
       property output : {}
   end script
   
   set listLen to (count o's lst)
   repeat with i from 1 to listLen by sublistLen
       set j to i + sublistLen - 1
       if (j > listLen) then set j to listLen
       set end of o's output to items i thru j of o's lst
   end repeat
   return o's output

end chopList

on matrixToText(matrix, w)

   script o
       property matrix : missing value
       property row : missing value
   end script
   
   set o's matrix to matrix
   set padding to "                   "
   repeat with r from 1 to (count o's matrix)
       set o's row to o's matrix's item r
       repeat with i from 1 to (count o's row)
           set o's row's item i to text -w thru end of (padding & o's row's item i)
       end repeat
       set o's matrix's item r to join(o's row, "")
   end repeat
   
   return join(o's matrix, linefeed)

end matrixToText

on join(lst, delim)

   set astid to AppleScript's text item delimiters
   set AppleScript's text item delimiters to delim
   set txt to lst as text
   set AppleScript's text item delimiters to astid
   return txt

end join

on task()

   set output to {"First 30 Jacobsthal Numbers:", "First 30 Jacobsthal-Lucas Numbers:", ¬
       "First 20 Jacobsthal oblong Numbers:", "First 11 Jacobsthal Primes:"}
   set results to {jacobsthalNumbers("", 30), jacobsthalNumbers("Lucas", 30), ¬
       jacobsthalNumbers("oblong", 20), jacobsthalNumbers("prime", 11)}
   repeat with i from 1 to 4
       set thisSequence to item i of results
       repeat with j in thisSequence
           set j's contents to intToText(j)
       end repeat
       if (i < 4) then
           set theLines to chopList(thisSequence, 10)
       else
           set theLines to chopList(thisSequence, 6)
       end if
       set item i of output to item i of output & linefeed & matrixToText(theLines, (count end of thisSequence) + 1)
   end repeat
   
   return join(output, linefeed & linefeed)

end task

task() </lang>

Output:

<lang applescript>"First 30 Jacobsthal Numbers:

        0         1         1         3         5        11        21        43        85       171
      341       683      1365      2731      5461     10923     21845     43691     87381    174763
   349525    699051   1398101   2796203   5592405  11184811  22369621  44739243  89478485 178956971

First 30 Jacobsthal-Lucas Numbers:

        2         1         5         7        17        31        65       127       257       511
     1025      2047      4097      8191     16385     32767     65537    131071    262145    524287
  1048577   2097151   4194305   8388607  16777217  33554431  67108865 134217727 268435457 536870911

First 20 Jacobsthal oblong Numbers:

          0           1           3          15          55         231         903        3655       14535       58311
     232903      932295     3727815    14913991    59650503   238612935   954429895  3817763271 15270965703 61084037575

First 11 Jacobsthal Primes:

            3             5            11            43           683          2731
        43691        174763       2796203     715827883 2932031007403"</lang>

Functional

<lang applescript>-------------------- JACOBSTHAL NUMBERS ------------------

-- e.g. take(10, jacobsthal())

-- jacobsthal :: [Int] on jacobsthal()

   -- The terms of OEIS:A001045 as a non-finite sequence.
   jacobsthalish(0, 1)

end jacobsthal

-- jacobsthal :: (Int, Int) -> [Int] on jacobsthalish(x, y)

   -- An infinite sequence of the terms of the 
   -- Jacobsthal-type series which begins with x and y.
   script go
       on |λ|(ab)
           set {a, b} to ab
           
           {a, {b, (2 * a) + b}}
       end |λ|
   end script
   
   unfoldr(go, {x, y})

end jacobsthalish

TESTS -------------------------

on run

   unlines(map(fShow, {¬
       {"terms of the Jacobsthal sequence", ¬
           30, jacobsthal()}, ¬
       {"Jacobsthal-Lucas numbers", ¬
           30, jacobsthalish(2, 1)}, ¬
       {"Jacobsthal oblong numbers", ¬
           20, zipWith(my mul, jacobsthal(), drop(1, jacobsthal()))}, ¬
       {"primes in the Jacobsthal sequence", ¬
           10, filter(isPrime, jacobsthal())}}))

end run

FORMATTING ----------------------

on fShow(test)

   set {k, n, xs} to test
   
   str(n) & " first " & k & ":" & linefeed & ¬
       table(5, map(my str, take(n, xs))) & linefeed

end fShow

-- justifyRight :: Int -> Char -> String -> String on justifyRight(n, cFiller)

   script go
       on |λ|(s)
           if n > length of s then
               text -n thru -1 of ((replicate(n, cFiller) as text) & s)
           else
               s
           end if
       end |λ|
   end script

end justifyRight

-- Egyptian multiplication - progressively doubling a list, appending -- stages of doubling to an accumulator where needed for binary -- assembly of a target length -- replicate :: Int -> String -> String on replicate(n, s)

   -- Egyptian multiplication - progressively doubling a list, 
   -- appending stages of doubling to an accumulator where needed 
   -- for binary assembly of a target length
   script p
       on |λ|({n})
           n ≤ 1
       end |λ|
   end script
   
   script f
       on |λ|({n, dbl, out})
           if (n mod 2) > 0 then
               set d to out & dbl
           else
               set d to out
           end if
           {n div 2, dbl & dbl, d}
       end |λ|
   end script
   
   set xs to |until|(p, f, {n, s, ""})
   item 2 of xs & item 3 of xs

end replicate

-- table :: Int -> [String] -> String on table(n, xs)

   -- A list of strings formatted as
   -- right-justified rows of n columns.
   set w to length of last item of xs
   unlines(map(my unwords, ¬
       chunksOf(n, map(justifyRight(w, space), xs))))

end table

-- unlines :: [String] -> String on unlines(xs)

   -- A single string formed by the intercalation
   -- of a list of strings with the newline character.
   set {dlm, my text item delimiters} to ¬
       {my text item delimiters, linefeed}
   set s to xs as text
   set my text item delimiters to dlm
   s

end unlines

-- until :: (a -> Bool) -> (a -> a) -> a -> a on |until|(p, f, x)

   set v to x
   set mp to mReturn(p)
   set mf to mReturn(f)
   repeat until mp's |λ|(v)
       set v to mf's |λ|(v)
   end repeat
   v

end |until|

-- unwords :: [String] -> String on unwords(xs)

   set {dlm, my text item delimiters} to ¬
       {my text item delimiters, space}
   set s to xs as text
   set my text item delimiters to dlm
   return s

end unwords

GENERIC ------------------------

-- Just :: a -> Maybe a on Just(x)

   -- Constructor for an inhabited Maybe (option type) value.
   -- Wrapper containing the result of a computation.
   {type:"Maybe", Nothing:false, Just:x}

end Just

-- Nothing :: Maybe a on Nothing()

   -- Constructor for an empty Maybe (option type) value.
   -- Empty wrapper returned where a computation is not possible.
   {type:"Maybe", Nothing:true}

end Nothing

-- abs :: Num -> Num on abs(x)

   -- Absolute value.
   if 0 > x then
       -x
   else
       x
   end if

end abs

-- any :: (a -> Bool) -> [a] -> Bool on any(p, xs)

   -- Applied to a predicate and a list, 
   -- |any| returns true if at least one element of the 
   -- list satisfies the predicate.
   tell mReturn(p)
       set lng to length of xs
       repeat with i from 1 to lng
           if |λ|(item i of xs) then return true
       end repeat
       false
   end tell

end any

-- chunksOf :: Int -> [a] -> a on chunksOf(k, xs)

   script
       on go(ys)
           set ab to splitAt(k, ys)
           set a to item 1 of ab
           if {} ≠ a then
               {a} & go(item 2 of ab)
           else
               a
           end if
       end go
   end script
   result's go(xs)

end chunksOf

-- drop :: Int -> [a] -> [a] -- drop :: Int -> String -> String on drop(n, xs)

   take(n, xs) -- consumed
   xs

end drop

-- enumFromThenTo :: Int -> Int -> Int -> [Int] on enumFromThenTo(x1, x2, y)

   set xs to {}
   set gap to x2 - x1
   set d to max(1, abs(gap)) * (signum(gap))
   repeat with i from x1 to y by d
       set end of xs to i
   end repeat
   return xs

end enumFromThenTo

-- filter :: (a -> Bool) -> Gen [a] -> Gen [a] on filter(p, gen)

   -- Non-finite stream of values which are 
   -- drawn from gen, and satisfy p
   script
       property mp : mReturn(p)'s |λ|
       on |λ|()
           set v to gen's |λ|()
           repeat until mp(v)
               set v to gen's |λ|()
           end repeat
           return v
       end |λ|
   end script

end filter

-- isPrime :: Int -> Bool on isPrime(n)

   -- True if n is prime
   
   if {2, 3} contains n then return true
   
   if 2 > n or 0 = (n mod 2) then return false
   
   if 9 > n then return true
   
   if 0 = (n mod 3) then return false
   
   script p
       on |λ|(x)
           0 = n mod x or 0 = n mod (2 + x)
       end |λ|
   end script
   
   not any(p, enumFromThenTo(5, 11, 1 + (n ^ 0.5)))

end isPrime

-- length :: [a] -> Int on |length|(xs)

   set c to class of xs
   if list is c or string is c then
       length of xs
   else
       (2 ^ 29 - 1) -- (maxInt - simple proxy for non-finite)
   end if

end |length|

-- map :: (a -> b) -> [a] -> [b] on map(f, xs)

   -- The list obtained by applying f
   -- to each element of xs.
   tell mReturn(f)
       set lng to length of xs
       set lst to {}
       repeat with i from 1 to lng
           set end of lst to |λ|(item i of xs, i, xs)
       end repeat
       return lst
   end tell

end map

-- max :: Ord a => a -> a -> a on max(x, y)

   if x > y then
       x
   else
       y
   end if

end max

-- mReturn :: First-class m => (a -> b) -> m (a -> b) on mReturn(f)

   -- 2nd class handler function lifted into 1st class script wrapper. 
   if script is class of f then
       f
   else
       script
           property |λ| : f
       end script
   end if

end mReturn

-- mul (*) :: Num a => a -> a -> a on mul(a, b)

   a * b

end mul

-- signum :: Num -> Num on signum(x)

   if x < 0 then
       -1
   else if x = 0 then
       0
   else
       1
   end if

end signum

-- splitAt :: Int -> [a] -> ([a], [a]) on splitAt(n, xs)

   if n > 0 and n < length of xs then
       if class of xs is text then
           {items 1 thru n of xs as text, ¬
               items (n + 1) thru -1 of xs as text}
       else
           {items 1 thru n of xs, items (n + 1) thru -1 of xs}
       end if
   else
       if n < 1 then
           {{}, xs}
       else
           {xs, {}}
       end if
   end if

end splitAt

-- str :: a -> String on str(x)

   x as string

end str

-- take :: Int -> [a] -> [a] -- take :: Int -> String -> String on take(n, xs)

   set ys to {}
   repeat with i from 1 to n
       set v to |λ|() of xs
       if missing value is v then
           return ys
       else
           set end of ys to v
       end if
   end repeat
   return ys

end take

-- uncons :: [a] -> Maybe (a, [a]) on uncons(xs)

   set lng to |length|(xs)
   if 0 = lng then
       Nothing()
   else
       if (2 ^ 29 - 1) as integer > lng then
           if class of xs is string then
               set cs to text items of xs
               Just({item 1 of cs, rest of cs})
           else
               Just({item 1 of xs, rest of xs})
           end if
       else
           set nxt to take(1, xs)
           if {} is nxt then
               Nothing()
           else
               Just({item 1 of nxt, xs})
           end if
       end if
   end if

end uncons

-- unfoldr :: (b -> Maybe (a, b)) -> b -> [a] on unfoldr(f, v)

   -- A lazy (generator) list unfolded from a seed value
   -- by repeated application of f to a value until no
   -- residue remains. Dual to fold/reduce.
   -- f returns either nothing (missing value) 
   -- or just (value, residue).
   script
       property valueResidue : {v, v}
       property g : mReturn(f)
       on |λ|()
           set valueResidue to g's |λ|(item 2 of (valueResidue))
           if missing value ≠ valueResidue then
               item 1 of (valueResidue)
           else
               missing value
           end if
       end |λ|
   end script

end unfoldr

-- zipWith :: (a -> b -> c) -> Gen [a] -> Gen [b] -> Gen [c] on zipWith(f, ga, gb)

   script
       property ma : missing value
       property mb : missing value
       property mf : mReturn(f)
       on |λ|()
           if missing value is ma then
               set ma to uncons(ga)
               set mb to uncons(gb)
           end if
           if Nothing of ma or Nothing of mb then
               missing value
           else
               set ta to Just of ma
               set tb to Just of mb
               set ma to uncons(item 2 of ta)
               set mb to uncons(item 2 of tb)
               |λ|(item 1 of ta, item 1 of tb) of mf
           end if
       end |λ|
   end script

end zipWith</lang>

Output:

30 first terms of the Jacobsthal sequence:
        0         1         1         3         5
       11        21        43        85       171
      341       683      1365      2731      5461
    10923     21845     43691     87381    174763
   349525    699051   1398101   2796203   5592405
 11184811  22369621  44739243  89478485 178956971

30 first Jacobsthal-Lucas numbers:
        2         1         5         7        17
       31        65       127       257       511
     1025      2047      4097      8191     16385
    32767     65537    131071    262145    524287
  1048577   2097151   4194305   8388607  16777217
 33554431  67108865 134217727 268435457 536870911

20 first Jacobsthal oblong numbers:
               0                1                3               15               55
             231              903             3655            14535            58311
          232903           932295          3727815         14913991         59650503
       238612935    9.54429895E+8   3.817763271E+9 1.5270965703E+10 6.1084037575E+10

10 first primes in the Jacobsthal sequence:
            3             5            11            43           683
         2731         43691        174763       2796203 7.15827883E+8

C

Library: GMP

<lang c>#include <stdio.h>

include <gmp.h>

void jacobsthal(mpz_t r, unsigned long n) {

   mpz_t s;
   mpz_init(s);
   mpz_set_ui(r, 1);
   mpz_mul_2exp(r, r, n);
   mpz_set_ui(s, 1);
   if (n % 2) mpz_neg(s, s);
   mpz_sub(r, r, s);
   mpz_div_ui(r, r, 3);

}

void jacobsthal_lucas(mpz_t r, unsigned long n) {

   mpz_t a;
   mpz_init(a);
   mpz_set_ui(r, 1);
   mpz_mul_2exp(r, r, n);
   mpz_set_ui(a, 1);
   if (n % 2) mpz_neg(a, a);
   mpz_add(r, r, a);

}

int main() {

   int i, count;
   mpz_t jac[30], j;
   printf("First 30 Jacobsthal numbers:\n");
   for (i = 0; i < 30; ++i) {
       mpz_init(jac[i]);
       jacobsthal(jac[i], i);
       gmp_printf("%9Zd ", jac[i]);
       if (!((i+1)%5)) printf("\n");
   }

   printf("\nFirst 30 Jacobsthal-Lucas numbers:\n");
   mpz_init(j);
   for (i = 0; i < 30; ++i) {
       jacobsthal_lucas(j, i);
       gmp_printf("%9Zd ", j);
       if (!((i+1)%5)) printf("\n");
   }

   printf("\nFirst 20 Jacobsthal oblong numbers:\n");
   for (i = 0; i < 20; ++i) {
       mpz_mul(j, jac[i], jac[i+1]);
       gmp_printf("%11Zd ", j);
       if (!((i+1)%5)) printf("\n");
   }

   printf("\nFirst 20 Jacobsthal primes:\n");
   for (i = 0, count = 0; count < 20; ++i) {
       jacobsthal(j, i);
       if (mpz_probab_prime_p(j, 15) > 0) {
           gmp_printf("%Zd\n", j);
           ++count;
       }
   }

   return 0;

}</lang>

Output:

First 30 Jacobsthal numbers:
        0         1         1         3         5 
       11        21        43        85       171 
      341       683      1365      2731      5461 
    10923     21845     43691     87381    174763 
   349525    699051   1398101   2796203   5592405 
 11184811  22369621  44739243  89478485 178956971 

First 30 Jacobsthal-Lucas numbers:
        2         1         5         7        17 
       31        65       127       257       511 
     1025      2047      4097      8191     16385 
    32767     65537    131071    262145    524287 
  1048577   2097151   4194305   8388607  16777217 
 33554431  67108865 134217727 268435457 536870911 

First 20 Jacobsthal oblong numbers:
          0           1           3          15          55 
        231         903        3655       14535       58311 
     232903      932295     3727815    14913991    59650503 
  238612935   954429895  3817763271 15270965703 61084037575 

First 20 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403
768614336404564651
201487636602438195784363
845100400152152934331135470251
56713727820156410577229101238628035243
62357403192785191176690552862561408838653121833643
1046183622564446793972631570534611069350392574077339085483
267823007376498379256993682056860433753700498963798805883563
5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443

C++

Library: GMP

<lang cpp>#include <gmpxx.h>

include <iomanip>
include <iostream>

using big_int = mpz_class;

bool is_probably_prime(const big_int& n) {

   return mpz_probab_prime_p(n.get_mpz_t(), 30) != 0;

}

big_int jacobsthal_number(unsigned int n) {

   return ((big_int(1) << n) - (n % 2 == 0 ? 1 : -1)) / 3;

}

big_int jacobsthal_lucas_number(unsigned int n) {

   return (big_int(1) << n) + (n % 2 == 0 ? 1 : -1);

}

big_int jacobsthal_oblong_number(unsigned int n) {

   return jacobsthal_number(n) * jacobsthal_number(n + 1);

}

int main() {

   std::cout << "First 30 Jacobsthal Numbers:\n";
   for (unsigned int n = 0; n < 30; ++n) {
       std::cout << std::setw(9) << jacobsthal_number(n)
                 << ((n + 1) % 5 == 0 ? '\n' : ' ');
   }
   std::cout << "\nFirst 30 Jacobsthal-Lucas Numbers:\n";
   for (unsigned int n = 0; n < 30; ++n) {
       std::cout << std::setw(9) << jacobsthal_lucas_number(n)
                 << ((n + 1) % 5 == 0 ? '\n' : ' ');
   }
   std::cout << "\nFirst 20 Jacobsthal oblong Numbers:\n";
   for (unsigned int n = 0; n < 20; ++n) {
       std::cout << std::setw(11) << jacobsthal_oblong_number(n)
                 << ((n + 1) % 5 == 0 ? '\n' : ' ');
   }
   std::cout << "\nFirst 20 Jacobsthal primes:\n";
   for (unsigned int n = 0, count = 0; count < 20; ++n) {
       auto jn = jacobsthal_number(n);
       if (is_probably_prime(jn)) {
           ++count;
           std::cout << jn << '\n';
       }
   }

}</lang>

Output:

First 30 Jacobsthal Numbers:
        0         1         1         3         5
       11        21        43        85       171
      341       683      1365      2731      5461
    10923     21845     43691     87381    174763
   349525    699051   1398101   2796203   5592405
 11184811  22369621  44739243  89478485 178956971

First 30 Jacobsthal-Lucas Numbers:
        2         1         5         7        17
       31        65       127       257       511
     1025      2047      4097      8191     16385
    32767     65537    131071    262145    524287
  1048577   2097151   4194305   8388607  16777217
 33554431  67108865 134217727 268435457 536870911

First 20 Jacobsthal oblong Numbers:
          0           1           3          15          55
        231         903        3655       14535       58311
     232903      932295     3727815    14913991    59650503
  238612935   954429895  3817763271 15270965703 61084037575

First 20 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403
768614336404564651
201487636602438195784363
845100400152152934331135470251
56713727820156410577229101238628035243
62357403192785191176690552862561408838653121833643
1046183622564446793972631570534611069350392574077339085483
267823007376498379256993682056860433753700498963798805883563
5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443

Factor

Works with: Factor version 0.99 2021-06-02

<lang factor>USING: grouping io kernel lists lists.lazy math math.functions math.primes prettyprint sequences ;

2^-1^ ( n -- 2^n -1^n ) dup 2^ -1 rot ^ ;

jacobsthal ( m -- n ) 2^-1^ - 3 / ;

jacobsthal-lucas ( m -- n ) 2^-1^ + ;

as-list ( quot -- list ) 0 lfrom swap lmap-lazy ; inline

jacobsthals ( -- list ) [ jacobsthal ] as-list ;

lucas-jacobthals ( -- list ) [ jacobsthal-lucas ] as-list ;

prime-jacobsthals ( -- list ) jacobsthals [ prime? ] lfilter ;

show ( n list -- ) ltake list>array 5 group simple-table. nl ;

oblong ( -- list )

   jacobsthals dup cdr lzip [ product ] lmap-lazy ;

"First 30 Jacobsthal numbers:" print 30 jacobsthals show

"First 30 Jacobsthal-Lucas numbers:" print 30 lucas-jacobthals show

"First 20 Jacobsthal oblong numbers:" print 20 oblong show

"First 20 Jacobsthal primes:" print 20 prime-jacobsthals ltake [ . ] leach</lang>

Output:

First 30 Jacobsthal numbers:
0        1        1        3        5
11       21       43       85       171
341      683      1365     2731     5461
10923    21845    43691    87381    174763
349525   699051   1398101  2796203  5592405
11184811 22369621 44739243 89478485 178956971

First 30 Jacobsthal-Lucas numbers:
2        1        5         7         17
31       65       127       257       511
1025     2047     4097      8191      16385
32767    65537    131071    262145    524287
1048577  2097151  4194305   8388607   16777217
33554431 67108865 134217727 268435457 536870911

First 20 Jacobsthal oblong numbers:
0         1         3          15          55
231       903       3655       14535       58311
232903    932295    3727815    14913991    59650503
238612935 954429895 3817763271 15270965703 61084037575

First 20 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403
768614336404564651
201487636602438195784363
845100400152152934331135470251
56713727820156410577229101238628035243
62357403192785191176690552862561408838653121833643
1046183622564446793972631570534611069350392574077339085483
267823007376498379256993682056860433753700498963798805883563
5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443

FreeBASIC

<lang freebasic>Function isPrime(n As Ulongint) As Boolean

   If n < 2 Then Return False
   If n Mod 2 = 0 Then Return false
   For i As Uinteger = 3 To Int(Sqr(n))+1 Step 2
       If n Mod i = 0 Then Return false
   Next i
   Return true

End Function

Dim Shared As Uinteger n(1) Dim Shared As Uinteger i0 = 0, i1 = 1 Dim Shared As Integer j, c, P = 1, Q = -2

Print "First 30 Jacobsthal numbers:" c = 0 : n(i0) = 0: n(i1) = 1 For j = 0 To 29

   c += 1
   Print Using " #########"; n(i0);
   Print Iif (c Mod 5, "", !"\n");
   n(i0) = P * n(i1) - Q * n(i0)
   Swap i0, i1

Next j

Print !"\n\nFirst 30 Jacobsthal-Lucas numbers: " c = 0 : n(i0) = 2: n(i1) = 1 For j = 0 To 29

   c += 1
   Print Using " #########"; n(i0);
   Print Iif (c Mod 5, "", !"\n");
   n(i0) = P * n(i1) - Q * n(i0)
   Swap i0, i1

Next j

Print !"\n\nFirst 20 Jacobsthal oblong numbers: " c = 0 : n(i0) = 0: n(i1) = 1 For j = 0 To 19

   c += 1
   Print Using " ###########"; n(i0)*n(i1);
   Print Iif (c Mod 5, "", !"\n");
   n(i0) = P * n(i1) - Q * n(i0)
   Swap i0, i1

Next j

Print !"\n\nFirst 10 Jacobsthal primes: " c = 0 : n(i0) = 0: n(i1) = 1 Do

   If isPrime(n(i0)) Then c += 1 : Print n(i0)
   n(i0) = P * n(i1) - Q * n(i0)
   Swap i0, i1

Loop Until c = 10 Sleep</lang>

Output:

First 30 Jacobsthal numbers:
        0         1         1         3         5
       11        21        43        85       171
      341       683      1365      2731      5461
    10923     21845     43691     87381    174763
   349525    699051   1398101   2796203   5592405
 11184811  22369621  44739243  89478485 178956971

First 30 Jacobsthal-Lucas numbers:
        2         1         5         7        17
       31        65       127       257       511
     1025      2047      4097      8191     16385
    32767     65537    131071    262145    524287
  1048577   2097151   4194305   8388607  16777217
 33554431  67108865 134217727 268435457 536870911

First 20 Jacobsthal oblong numbers:
          0           1           3          15          55
        231         903        3655       14535       58311
     232903      932295     3727815    14913991    59650503
  238612935   954429895  3817763271 15270965703 61084037575

First 10 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883

Go

<lang go>package main

import (

   "fmt"
   "math/big"

)

func jacobsthal(n uint) *big.Int {

   t := big.NewInt(1)
   t.Lsh(t, n)
   s := big.NewInt(1)
   if n%2 != 0 {
       s.Neg(s)
   }
   t.Sub(t, s)
   return t.Div(t, big.NewInt(3))

}

func jacobsthalLucas(n uint) *big.Int {

   t := big.NewInt(1)
   t.Lsh(t, n)
   a := big.NewInt(1)
   if n%2 != 0 {
       a.Neg(a)
   }
   return t.Add(t, a)

}

func main() {

   jac := make([]*big.Int, 30)
   fmt.Println("First 30 Jacobsthal numbers:")
   for i := uint(0); i < 30; i++ {
       jac[i] = jacobsthal(i)
       fmt.Printf("%9d ", jac[i])
       if (i+1)%5 == 0 {
           fmt.Println()
       }
   }

   fmt.Println("\nFirst 30 Jacobsthal-Lucas numbers:")
   for i := uint(0); i < 30; i++ {
       fmt.Printf("%9d ", jacobsthalLucas(i))
       if (i+1)%5 == 0 {
           fmt.Println()
       }
   }

   fmt.Println("\nFirst 20 Jacobsthal oblong numbers:")
   for i := uint(0); i < 20; i++ {
       t := big.NewInt(0)
       fmt.Printf("%11d ", t.Mul(jac[i], jac[i+1]))
       if (i+1)%5 == 0 {
           fmt.Println()
       }
   }

   fmt.Println("\nFirst 20 Jacobsthal primes:")
   for n, count := uint(0), 0; count < 20; n++ {
       j := jacobsthal(n)
       if j.ProbablyPrime(10) {
           fmt.Println(j)
           count++
       }
   }

}</lang>

Output:

First 30 Jacobsthal numbers:
        0         1         1         3         5 
       11        21        43        85       171 
      341       683      1365      2731      5461 
    10923     21845     43691     87381    174763 
   349525    699051   1398101   2796203   5592405 
 11184811  22369621  44739243  89478485 178956971 

First 30 Jacobsthal-Lucas numbers:
        2         1         5         7        17 
       31        65       127       257       511 
     1025      2047      4097      8191     16385 
    32767     65537    131071    262145    524287 
  1048577   2097151   4194305   8388607  16777217 
 33554431  67108865 134217727 268435457 536870911 

First 20 Jacobsthal oblong numbers:
          0           1           3          15          55 
        231         903        3655       14535       58311 
     232903      932295     3727815    14913991    59650503 
  238612935   954429895  3817763271 15270965703 61084037575 

First 20 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403
768614336404564651
201487636602438195784363
845100400152152934331135470251
56713727820156410577229101238628035243
62357403192785191176690552862561408838653121833643
1046183622564446793972631570534611069350392574077339085483
267823007376498379256993682056860433753700498963798805883563
5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443

Haskell

<lang haskell>jacobsthal :: [Integer] jacobsthal = 0 : 1 : zipWith (\x y -> 2 * x + y) jacobsthal (tail jacobsthal)

jacobsthalLucas :: [Integer] jacobsthalLucas = 2 : 1 : zipWith (\x y -> 2 * x + y) jacobsthalLucas (tail jacobsthalLucas)

jacobsthalOblong :: [Integer] jacobsthalOblong = zipWith (*) jacobsthal (tail jacobsthal)

isPrime :: Integer -> Bool isPrime n = n > 1 && not (or [n `mod` i == 0 | i <- [2 .. floor (sqrt (fromInteger n))]])

main :: IO () main = do

 putStrLn "First 30 Jacobsthal numbers:"
 print $ take 30 jacobsthal
 putStrLn ""
 putStrLn "First 30 Jacobsthal-Lucas numbers:"
 print $ take 30 jacobsthalLucas
 putStrLn ""
 putStrLn "First 20 Jacobsthal oblong numbers:"
 print $ take 20 jacobsthalOblong
 putStrLn ""
 putStrLn "First 10 Jacobsthal primes:"
 print $ take 10 $ filter isPrime jacobsthal</lang>

Output:

First 30 Jacobsthal numbers:
[0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691,87381,174763,349525,699051,1398101,2796203,5592405,11184811,22369621,44739243,89478485,178956971]

First 30 Jacobsthal-Lucas numbers:
[2,1,5,7,17,31,65,127,257,511,1025,2047,4097,8191,16385,32767,65537,131071,262145,524287,1048577,2097151,4194305,8388607,16777217,33554431,67108865,134217727,268435457,536870911]

First 20 Jacobsthal oblong numbers:
[0,1,3,15,55,231,903,3655,14535,58311,232903,932295,3727815,14913991,59650503,238612935,954429895,3817763271,15270965703,61084037575]

First 10 Jacobsthal primes:
[3,5,11,43,683,2731,43691,174763,2796203,715827883]

or, defined in terms of unfoldr:

<lang haskell>import Data.List (intercalate, transpose, uncons, unfoldr) import Data.List.Split (chunksOf) import Data.Numbers.Primes (isPrime) import Text.Printf (printf)

JACOBSTHAL NUMBERS ------------------

jacobsthal :: [Integer] jacobsthal = jacobsthalish (0, 1)

jacobsthalish :: (Integer, Integer) -> [Integer] jacobsthalish = unfoldr go

 where
   go (a, b) = Just (a, (b, 2 * a + b))

TEST -------------------------

main :: IO () main =

 mapM_
   (putStrLn . format)
   [ ( "terms of the Jacobsthal sequence",
       30,
       jacobsthal
     ),
     ( "Jacobsthal-Lucas numbers",
       30,
       jacobsthalish (2, 1)
     ),
     ( "Jacobsthal oblong numbers",
       20,
       zipWith (*) jacobsthal (tail jacobsthal)
     ),
     ( "Jacobsthal primes",
       10,
       filter isPrime jacobsthal
     )
   ]

format :: (String, Int, [Integer]) -> String format (k, n, xs) =

 show n <> (' ' : k) <> ":\n"
   <> table
     "  "
     (chunksOf 5 $ show <$> take n xs)

table :: String -> String -> String table gap rows =

 let ws = maximum . fmap length <$> transpose rows
     pw = printf . flip intercalate ["%", "s"] . show
  in unlines $ intercalate gap . zipWith pw ws <$> rows</lang>

Output:

30 terms of the Jacobsthal sequence:
       0         1         1         3          5
      11        21        43        85        171
     341       683      1365      2731       5461
   10923     21845     43691     87381     174763
  349525    699051   1398101   2796203    5592405
11184811  22369621  44739243  89478485  178956971

30 Jacobsthal-Lucas numbers:
       2         1          5          7         17
      31        65        127        257        511
    1025      2047       4097       8191      16385
   32767     65537     131071     262145     524287
 1048577   2097151    4194305    8388607   16777217
33554431  67108865  134217727  268435457  536870911

20 Jacobsthal oblong numbers:
        0          1           3           15           55
      231        903        3655        14535        58311
   232903     932295     3727815     14913991     59650503
238612935  954429895  3817763271  15270965703  61084037575

10 Jacobsthal primes:
   3      5      11       43        683
2731  43691  174763  2796203  715827883

J

Implementation:

<lang J>ja=: 3 %~ 2x&^ - _1x&^ NB. Jacobsthal jl=: 2x&^ + _1x&^ NB.Jacobsthal-Lucas</lang>

Task examples:

<lang J> ja i.3 10

    0      1       1       3       5       11       21       43       85       171
  341    683    1365    2731    5461    10923    21845    43691    87381    174763

349525 699051 1398101 2796203 5592405 11184811 22369621 44739243 89478485 178956971

  jl i.3 10
     2       1       5       7       17       31       65       127       257       511
  1025    2047    4097    8191    16385    32767    65537    131071    262145    524287

1048577 2097151 4194305 8388607 16777217 33554431 67108865 134217727 268435457 536870911

  2 10$2 */\ ja i.21 NB. Jacobsthal oblong
    0      1       3       15       55       231       903       3655       14535       58311

232903 932295 3727815 14913991 59650503 238612935 954429895 3817763271 15270965703 61084037575

  ja I.1 p:ja i.32  NB. first ten Jacobsthal primes

3 5 11 43 683 2731 43691 174763 2796203 715827883</lang>

jq

Works with gojq, the Go implementation of jq

See Erdős-primes#jq for a suitable definition of `is_prime` as used here. As a practical matter, this function limits the exploration of Jacobsthal primes.

Preliminaries <lang jq># Split the input array into a stream of arrays def chunks(n):

 def c: .[0:n], (if length > n then .[n:]|c else empty end);
 c;

def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;

If $j is 0, then an error condition is raised;
otherwise, assuming infinite-precision integer arithmetic,
if the input and $j are integers, then the result will be a pair of integers.

def divmod($j):

 . as $i
 | ($i % $j) as $mod
 | [($i - $mod) / $j, $mod] ;

To take advantage of gojq's arbitrary-precision integer arithmetic:

def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);</lang>

The Tasks <lang jq>def jacobsthal:

 . as $n
 | ( (2|power($n)) - (if ($n%2 == 0) then 1 else -1 end)) | divmod(3)[0];

def jacobsthalLucas:

 . as $n
 | (2|power($n)) + (if ($n%2 == 0) then 1 else -1 end);

def tasks:

 def pp($width): chunks(5) | map(lpad($width)) | join("");
 
 [range(0;30) | jacobsthal] as $js
 | "First 30 Jacobsthal numbers:",
   ( $js | pp(12)),

   "\nFirst 30 Jacobsthal-Lucas numbers:",
   ( [range(0;30) | jacobsthalLucas]  | pp(12)),

   "\nFirst 20 Jacobsthal oblong numbers:",
   ( [range(0;20) | $js[.] * $js[1+.]] | pp(14)),

  "\nFirst 11 Jacobsthal primes:",
   limit(11; range(0; infinite) | jacobsthal | select(is_prime))

tasks</lang>

Output:

First 30 Jacobsthal numbers:
           0           1           1           3           5
          11          21          43          85         171
         341         683        1365        2731        5461
       10923       21845       43691       87381      174763
      349525      699051     1398101     2796203     5592405
    11184811    22369621    44739243    89478485   178956971

First 30 Jacobsthal-Lucas numbers:
           2           1           5           7          17
          31          65         127         257         511
        1025        2047        4097        8191       16385
       32767       65537      131071      262145      524287
     1048577     2097151     4194305     8388607    16777217
    33554431    67108865   134217727   268435457   536870911

First 20 Jacobsthal oblong numbers:
             0             1             3            15            55
           231           903          3655         14535         58311
        232903        932295       3727815      14913991      59650503
     238612935     954429895    3817763271   15270965703   61084037575

First 11 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403

Julia

<lang julia>using Lazy using Primes

J(n) = (2^n - (-1)^n) ÷ 3 L(n) = 2^n + (-1)^n

Jacobsthal = @>> Lazy.range(0) map(J) JLucas = @>> Lazy.range(0) map(L) Joblong = @>> Lazy.range(big"0") map(n -> J(n) * J(n + 1)) Jprimes = @>> Lazy.range(big"0") map(J) filter(isprime)

function printrows(title, vec, columnsize = 15, columns = 5, rjust=true)

   println(title)
   for (i, n) in enumerate(vec)
       print((rjust ? lpad : rpad)(n, columnsize), i % columns == 0 ? "\n" : "")
   end
   println()

end

printrows("Thirty Jacobsthal numbers:", collect(take(30, Jacobsthal))) printrows("Thirty Jacobsthal-Lucas numbers:", collect(take(30, JLucas))) printrows("Twenty oblong Jacobsthal numbers:", collect(take(20, Joblong))) printrows("Fifteen Jacabsthal prime numbers:", collect(take(15, Jprimes)), 40, 1, false)

</lang>

Output:

Thirty Jacobsthal numbers:
              0              1              1              3              5
             11             21             43             85            171
            341            683           1365           2731           5461
          10923          21845          43691          87381         174763
         349525         699051        1398101        2796203        5592405
       11184811       22369621       44739243       89478485      178956971

Thirty Jacobsthal-Lucas numbers:
              2              1              5              7             17
             31             65            127            257            511
           1025           2047           4097           8191          16385
          32767          65537         131071         262145         524287
        1048577        2097151        4194305        8388607       16777217
       33554431       67108865      134217727      268435457      536870911

Twenty oblong Jacobsthal numbers:
              0              1              3             15             55
            231            903           3655          14535          58311
         232903         932295        3727815       14913991       59650503
      238612935      954429895     3817763271    15270965703    61084037575

Fifteen Jacabsthal prime numbers:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403
768614336404564651
201487636602438195784363
845100400152152934331135470251
56713727820156410577229101238628035243

Perl

Library: ntheory

<lang perl>use strict; use warnings; use feature <say state>; use bigint; use List::Util 'max'; use ntheory 'is_prime';

sub table { my $t = 5 * (my $c = 1 + length max @_); ( sprintf( ('%'.$c.'d')x@_, @_) ) =~ s/.{1,$t}\K/\n/gr }

sub jacobsthal { my($n) = @_; state @J = (0, 1); do { push @J, $J[-1] + 2 * $J[-2]} until @J > $n; $J[$n] } sub jacobsthal_lucas { my($n) = @_; state @JL = (2, 1); do { push @JL, $JL[-1] + 2 * $JL[-2]} until @JL > $n; $JL[$n] }

my(@j,@jp,$c,$n); push @j, jacobsthal $_ for 0..29; do { is_prime($n = ( 2**++$c - -1**$c ) / 3) and push @jp, $n } until @jp == 20;

say "First 30 Jacobsthal numbers:\n", table @j; say "First 30 Jacobsthal-Lucas numbers:\n", table map { jacobsthal_lucas $_-1 } 1..30; say "First 20 Jacobsthal oblong numbers:\n", table map { $j[$_-1] * $j[$_] } 1..20; say "First 20 Jacobsthal primes:\n", join "\n", @jp;</lang>

Output:

First 30 Jacobsthal numbers:
         0         1         1         3         5
        11        21        43        85       171
       341       683      1365      2731      5461
     10923     21845     43691     87381    174763
    349525    699051   1398101   2796203   5592405
  11184811  22369621  44739243  89478485 178956971

First 30 Jacobsthal-Lucas numbers:
         2         1         5         7        17
        31        65       127       257       511
      1025      2047      4097      8191     16385
     32767     65537    131071    262145    524287
   1048577   2097151   4194305   8388607  16777217
  33554431  67108865 134217727 268435457 536870911

First 20 Jacobsthal oblong numbers:
           0           1           3          15          55
         231         903        3655       14535       58311
      232903      932295     3727815    14913991    59650503
   238612935   954429895  3817763271 15270965703 61084037575

First 20 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403
768614336404564651
201487636602438195784363
845100400152152934331135470251
56713727820156410577229101238628035243
62357403192785191176690552862561408838653121833643
1046183622564446793972631570534611069350392574077339085483
267823007376498379256993682056860433753700498963798805883563
5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443

Phix

You can run this online here.

with javascript_semantics 
function jacobsthal(integer n)
    return floor((power(2,n)+odd(n))/3)
end function

function jacobsthal_lucas(integer n)
    return power(2,n)+power(-1,n)
end function

function jacobsthal_oblong(integer n)
    return jacobsthal(n)*jacobsthal(n+1)
end function

printf(1,"First 30 Jacobsthal numbers:\n%s\n",       {join_by(apply(true,sprintf,{{"%9d" },apply(tagset(29,0),jacobsthal)}),1,5," ")})
printf(1,"First 30 Jacobsthal-Lucas numbers:\n%s\n", {join_by(apply(true,sprintf,{{"%9d" },apply(tagset(29,0),jacobsthal_lucas)}),1,5," ")})
printf(1,"First 20 Jacobsthal oblong numbers:\n%s\n",{join_by(apply(true,sprintf,{{"%11d"},apply(tagset(19,0),jacobsthal_oblong)}),1,5," ")})
--printf(1,"First 10 Jacobsthal primes:\n%s\n",    {join(apply(true,sprintf,{{"%d"},filter(apply(tagset(31,0),jacobsthal),is_prime)}),"\n")})
--hmm(""), fine, but to go further roll out gmp:
include mpfr.e
mpz z = mpz_init()
integer n = 1, found = 0
printf(1,"First 20 jacobsthal primes:\n")
while found<20 do
    mpz_ui_pow_ui(z,2,n)
    mpz_add_ui(z,z,odd(n))
    {} = mpz_fdiv_q_ui(z,z,3)
    if mpz_prime(z) then
        found += 1
        printf(1,"%s\n",{mpz_get_str(z)})
    end if
    n += 1
end while

Likewise should you want the three basic functions to go further they'll have to look much more like the C submission above.

Output:

First 30 Jacobsthal numbers:
        0         1         1         3         5
       11        21        43        85       171
      341       683      1365      2731      5461
    10923     21845     43691     87381    174763
   349525    699051   1398101   2796203   5592405
 11184811  22369621  44739243  89478485 178956971

First 30 Jacobsthal-Lucas numbers:
        2         1         5         7        17
       31        65       127       257       511
     1025      2047      4097      8191     16385
    32767     65537    131071    262145    524287
  1048577   2097151   4194305   8388607  16777217
 33554431  67108865 134217727 268435457 536870911

First 20 Jacobsthal oblong numbers:
          0           1           3          15          55
        231         903        3655       14535       58311
     232903      932295     3727815    14913991    59650503
  238612935   954429895  3817763271 15270965703 61084037575

First 20 jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403
768614336404564651
201487636602438195784363
845100400152152934331135470251
56713727820156410577229101238628035243
62357403192785191176690552862561408838653121833643
1046183622564446793972631570534611069350392574077339085483
267823007376498379256993682056860433753700498963798805883563
5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443

Python

Translation of: Phix

<lang python>#!/usr/bin/python from math import floor, pow

def isPrime(n):

   for i in range(2, int(n**0.5) + 1):
       if n % i == 0:
           return False        
   return True

def odd(n):

   return n and 1 != 0

def jacobsthal(n):

   return floor((pow(2,n)+odd(n))/3)

def jacobsthal_lucas(n):

   return int(pow(2,n)+pow(-1,n))

def jacobsthal_oblong(n):

   return jacobsthal(n)*jacobsthal(n+1)

if __name__ == '__main__':

   print("First 30 Jacobsthal numbers:")
   for j in range(0, 30):
       print(jacobsthal(j), end="  ")

   print("\n\nFirst 30 Jacobsthal-Lucas numbers: ")
   for j in range(0, 30):
       print(jacobsthal_lucas(j), end = '\t')

   print("\n\nFirst 20 Jacobsthal oblong numbers: ")
   for j in range(0, 20):
       print(jacobsthal_oblong(j), end="  ")

   print("\n\nFirst 10 Jacobsthal primes: ")
   for j in range(3, 33):
       if isPrime(jacobsthal(j)):
           print(jacobsthal(j))</lang>

Output:

First 30 Jacobsthal numbers:
0  1  1  3  5  11  21  43  85  171  341  683  1365  2731  5461  10923  21845  43691  87381  174763  349525  699051  1398101  2796203  5592405  11184811  22369621  44739243  89478485  178956971  

First 30 Jacobsthal-Lucas numbers: 
2	1	5	7	17	31	65	127	257	511	1025	2047	4097	8191	16385	32767	65537	131071	262145	524287	1048577	2097151	4194305	8388607	16777217	33554431	67108865	134217727	268435457	536870911	

First 20 Jacobsthal oblong numbers: 
0  1  3  15  55  231  903  3655  14535  58311  232903  932295  3727815  14913991  59650503  238612935  954429895  3817763271  15270965703  61084037575  

First 10 Jacobsthal primes: 
3
5
11
43
683
2731
43691
174763
2796203
715827883

Or, defining an infinite series in terms of a general unfoldr anamorphism:

<lang python>Jacobsthal numbers

from itertools import islice from operator import mul

jacobsthal :: [Integer]

def jacobsthal():

   Infinite sequence of terms of OEIS A001045
   
   return jacobsthalish(0, 1)

jacobsthalish :: (Int, Int) -> [Int]

def jacobsthalish(x, y):

   Infinite sequence of jacobsthal-type series
      beginning with a, b
   
   def go(ab):
       a, b = ab
       return a, (b, 2 * a + b)

   return unfoldr(go)((x, y))

------------------------- TEST -------------------------
main :: IO ()

def main():

   First 15 terms each n-step Fibonacci(n) series
      where n is drawn from [2..8]
   
   print('\n\n'.join([
       fShow(*x) for x in [
           (
               'terms of the Jacobsthal sequence',
               30, jacobsthal()),
           (
               'Jacobsthal-Lucas numbers',
               30, jacobsthalish(2, 1)
           ),
           (
               'Jacobsthal oblong numbers',
               20, map(
                   mul, jacobsthal(),
                   drop(1)(jacobsthal())
               )
           ),
           (
               'primes in the Jacobsthal sequence',
               10, filter(isPrime, jacobsthal())
           )
       ]
   ]))

fShow :: (String, Int, [Integer]) -> String

def fShow(k, n, xs):

   N tabulated terms of XS, prefixed by the label K
   
   return f'{n} {k}:\n' + spacedTable(
       list(chunksOf(5)(
           [str(t) for t in take(n)(xs)]
       ))
   )

----------------------- GENERIC ------------------------

drop :: Int -> [a] -> [a]
drop :: Int -> String -> String

def drop(n):

   The sublist of xs beginning at
      (zero-based) index n.
   
   def go(xs):
       if isinstance(xs, (list, tuple, str)):
           return xs[n:]
       else:
           take(n)(xs)
           return xs
   return go

isPrime :: Int -> Bool

def isPrime(n):

   True if n is prime.
   if n in (2, 3):
       return True
   if 2 > n or 0 == n % 2:
       return False
   if 9 > n:
       return True
   if 0 == n % 3:
       return False

   def p(x):
       return 0 == n % x or 0 == n % (2 + x)

   return not any(map(p, range(5, 1 + int(n ** 0.5), 6)))

take :: Int -> [a] -> [a]
take :: Int -> String -> String

def take(n):

   The prefix of xs of length n,
      or xs itself if n > length xs.
   
   def go(xs):
       return (
           xs[0:n]
           if isinstance(xs, (list, tuple))
           else list(islice(xs, n))
       )
   return go

unfoldr :: (b -> Maybe (a, b)) -> b -> [a]

def unfoldr(f):

   Generic anamorphism.
      A lazy (generator) list unfolded from a seed value by
      repeated application of f until no residue remains.
      Dual to fold/reduce.
      f returns either None, or just (value, residue).
      For a strict output value, wrap in list().
   
   def go(x):
       valueResidue = f(x)
       while None is not valueResidue:
           yield valueResidue[0]
           valueResidue = f(valueResidue[1])
   return go

---------------------- FORMATTING ----------------------

chunksOf :: Int -> [a] -> a

def chunksOf(n):

   A series of lists of length n, subdividing the
      contents of xs. Where the length of xs is not evenly
      divisible, the final list will be shorter than n.
   
   def go(xs):
       return (
           xs[i:n + i] for i in range(0, len(xs), n)
       ) if 0 < n else None
   return go

spacedTable :: String -> String

def spacedTable(rows):

   Tabulated stringification of rows
   columnWidths = [
       max([len(x) for x in col])
       for col in zip(*rows)
   ]
   return '\n'.join([
       ' '.join(
           map(
               lambda x, w: x.rjust(w, ' '),
               row, columnWidths
           )
       )
       for row in rows
   ])

MAIN ---

if __name__ == '__main__':

   main()</lang>

Output:

30 terms of the Jacobsthal sequence:
       0        1        1        3         5
      11       21       43       85       171
     341      683     1365     2731      5461
   10923    21845    43691    87381    174763
  349525   699051  1398101  2796203   5592405
11184811 22369621 44739243 89478485 178956971

30 Jacobsthal-Lucas numbers:
       2        1         5         7        17
      31       65       127       257       511
    1025     2047      4097      8191     16385
   32767    65537    131071    262145    524287
 1048577  2097151   4194305   8388607  16777217
33554431 67108865 134217727 268435457 536870911

20 Jacobsthal oblong numbers:
        0         1          3          15          55
      231       903       3655       14535       58311
   232903    932295    3727815    14913991    59650503
238612935 954429895 3817763271 15270965703 61084037575

10 primes in the Jacobsthal sequence:
   3     5     11      43       683
2731 43691 174763 2796203 715827883

Raku

<lang perl6>my $jacobsthal = cache lazy 0, 1, * × 2 + * … *; my $jacobsthal-lucas = lazy 2, 1, * × 2 + * … *;

say "First 30 Jacobsthal numbers:"; say $jacobsthal[^30].batch(5)».fmt("%9d").join: "\n";

say "\nFirst 30 Jacobsthal-Lucas numbers:"; say $jacobsthal-lucas[^30].batch(5)».fmt("%9d").join: "\n";

say "\nFirst 20 Jacobsthal oblong numbers:"; say (^∞).map( { $jacobsthal[$_] × $jacobsthal[$_+1] } )[^20].batch(5)».fmt("%11d").join: "\n";

say "\nFirst 20 Jacobsthal primes:"; say $jacobsthal.grep( &is-prime )[^20].join: "\n";</lang>

Output:

First 30 Jacobsthal numbers:
        0         1         1         3         5
       11        21        43        85       171
      341       683      1365      2731      5461
    10923     21845     43691     87381    174763
   349525    699051   1398101   2796203   5592405
 11184811  22369621  44739243  89478485 178956971

First 30 Jacobsthal-Lucas numbers:
        2         1         5         7        17
       31        65       127       257       511
     1025      2047      4097      8191     16385
    32767     65537    131071    262145    524287
  1048577   2097151   4194305   8388607  16777217
 33554431  67108865 134217727 268435457 536870911

First 20 Jacobsthal oblong numbers:
          0           1           3          15          55
        231         903        3655       14535       58311
     232903      932295     3727815    14913991    59650503
  238612935   954429895  3817763271 15270965703 61084037575

First 20 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403
768614336404564651
201487636602438195784363
845100400152152934331135470251
56713727820156410577229101238628035243
62357403192785191176690552862561408838653121833643
1046183622564446793972631570534611069350392574077339085483
267823007376498379256993682056860433753700498963798805883563
5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443

Red

<lang rebol>Red ["Jacobsthal numbers"]

jacobsthal: function [n] [to-integer (2 ** n - (-1 ** n) / 3)]

lucas: function [n] [2 ** n + (-1 ** n)]

oblong: function [n] [

   first split mold multiply to-float jacobsthal n to-float jacobsthal n + 1 #"."   ; work around integer overflow

]

prime?: function [

   "Returns true if the input is a prime number"
   n [number!] "An integer to check for primality"

][

   if 2 = n [return true]
   if any [1 = n even? n] [return false]
   limit: sqrt n
   candidate: 3
   while [candidate < limit][
       if n % candidate = 0 [return false]
       candidate: candidate + 2
   ]
   true

]

show: function [n fn][

   cols: length? mold fn n
   repeat i n [
       prin [pad fn subtract i 1 cols]
       if i % 5 = 0 [prin newline]
   ]
   prin newline

]

print "First 30 Jacobsthal numbers:" show 30 :jacobsthal

print "First 30 Jacobsthal-Lucas numbers:" show 30 :lucas

print "First 20 Jacobsthal oblong numbers:" show 20 :oblong

print "First 10 Jacobsthal primes:" primes: n: 0 while [primes < 10][

   if prime? jacobsthal n [
       print jacobsthal n
       primes: primes + 1
   ]
   n: n + 1

]</lang>

Output:

First 30 Jacobsthal numbers:
0        1        1        3        5        
11       21       43       85       171      
341      683      1365     2731     5461     
10923    21845    43691    87381    174763   
349525   699051   1398101  2796203  5592405  
11184811 22369621 44739243 89478485 178956971

First 30 Jacobsthal-Lucas numbers:
2         1         5         7         17        
31        65        127       257       511       
1025      2047      4097      8191      16385     
32767     65537     131071    262145    524287    
1048577   2097151   4194305   8388607   16777217  
33554431  67108865  134217727 268435457 536870911 

First 20 Jacobsthal oblong numbers:
0             1             3             15            55            
231           903           3655          14535         58311         
232903        932295        3727815       14913991      59650503      
238612935     954429895     3817763271    15270965703   61084037575   

First 10 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883

Rust

<lang rust>// [dependencies] // rug = "0.3"

use rug::integer::IsPrime; use rug::Integer;

fn jacobsthal_numbers() -> impl std::iter::Iterator<Item = Integer> {

   (0..).map(|x| ((Integer::from(1) << x) - if x % 2 == 0 { 1 } else { -1 }) / 3)

}

fn jacobsthal_lucas_numbers() -> impl std::iter::Iterator<Item = Integer> {

   (0..).map(|x| (Integer::from(1) << x) + if x % 2 == 0 { 1 } else { -1 })

}

fn jacobsthal_oblong_numbers() -> impl std::iter::Iterator<Item = Integer> {

   let mut jn = jacobsthal_numbers();
   let mut n0 = jn.next().unwrap();
   std::iter::from_fn(move || {
       let n1 = jn.next().unwrap();
       let result = Integer::from(&n0 * &n1);
       n0 = n1;
       Some(result)
   })

}

fn jacobsthal_primes() -> impl std::iter::Iterator<Item = Integer> {

   jacobsthal_numbers().filter(|x| x.is_probably_prime(30) != IsPrime::No)

}

fn main() {

   println!("First 30 Jacobsthal Numbers:");
   for (i, n) in jacobsthal_numbers().take(30).enumerate() {
       print!("{:9}{}", n, if (i + 1) % 5 == 0 { "\n" } else { " " });
   }
   println!("\nFirst 30 Jacobsthal-Lucas Numbers:");
   for (i, n) in jacobsthal_lucas_numbers().take(30).enumerate() {
       print!("{:9}{}", n, if (i + 1) % 5 == 0 { "\n" } else { " " });
   }
   println!("\nFirst 20 Jacobsthal oblong Numbers:");
   for (i, n) in jacobsthal_oblong_numbers().take(20).enumerate() {
       print!("{:11}{}", n, if (i + 1) % 5 == 0 { "\n" } else { " " });
   }
   println!("\nFirst 20 Jacobsthal primes:");
   for n in jacobsthal_primes().take(20) {
       println!("{}", n);
   }

}</lang>

Output:

First 30 Jacobsthal Numbers:
        0         1         1         3         5
       11        21        43        85       171
      341       683      1365      2731      5461
    10923     21845     43691     87381    174763
   349525    699051   1398101   2796203   5592405
 11184811  22369621  44739243  89478485 178956971

First 30 Jacobsthal-Lucas Numbers:
        2         1         5         7        17
       31        65       127       257       511
     1025      2047      4097      8191     16385
    32767     65537    131071    262145    524287
  1048577   2097151   4194305   8388607  16777217
 33554431  67108865 134217727 268435457 536870911

First 20 Jacobsthal oblong Numbers:
          0           1           3          15          55
        231         903        3655       14535       58311
     232903      932295     3727815    14913991    59650503
  238612935   954429895  3817763271 15270965703 61084037575

First 20 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403
768614336404564651
201487636602438195784363
845100400152152934331135470251
56713727820156410577229101238628035243
62357403192785191176690552862561408838653121833643
1046183622564446793972631570534611069350392574077339085483
267823007376498379256993682056860433753700498963798805883563
5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443

Sidef

<lang ruby>func jacobsthal(n) {

   lucasU(1, -2, n)

}

func lucas_jacobsthal(n) {

   lucasV(1, -2, n)

}

say "First 30 Jacobsthal numbers:" say 30.of(jacobsthal)

say "\nFirst 30 Jacobsthal-Lucas numbers:" say 30.of(lucas_jacobsthal)

say "\nFirst 20 Jacobsthal oblong numbers:" say 21.of(jacobsthal).cons(2, {|a,b| a * b })

say "\nFirst 20 Jacobsthal primes:"; say (1..Inf -> lazy.map(jacobsthal).grep{.is_prime}.first(20))</lang>

Output:

First 30 Jacobsthal numbers:
[0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, 178956971]

First 30 Jacobsthal-Lucas numbers:
[2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, 2097151, 4194305, 8388607, 16777217, 33554431, 67108865, 134217727, 268435457, 536870911]

First 20 Jacobsthal oblong numbers:
[0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, 232903, 932295, 3727815, 14913991, 59650503, 238612935, 954429895, 3817763271, 15270965703, 61084037575]

First 20 Jacobsthal primes:
[3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, 62357403192785191176690552862561408838653121833643, 1046183622564446793972631570534611069350392574077339085483, 267823007376498379256993682056860433753700498963798805883563, 5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731, 95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443]

Wren

Library: Wren-big

Library: Wren-seq

Library: Wren-fmt

<lang ecmascript>import "./big" for BigInt import "./seq" for Lst import "./fmt" for Fmt

var jacobsthal = Fn.new { |n| ((BigInt.one << n) - ((n%2 == 0) ? 1 : -1)) / 3 }

var jacobsthalLucas = Fn.new { |n| (BigInt.one << n) + ((n%2 == 0) ? 1 : -1) }

System.print("First 30 Jacobsthal numbers:") var js = (0..29).map { |i| jacobsthal.call(i) }.toList for (chunk in Lst.chunks(js, 5)) Fmt.print("$,12i", chunk)

System.print("\nFirst 30 Jacobsthal-Lucas numbers:") var jsl = (0..29).map { |i| jacobsthalLucas.call(i) }.toList for (chunk in Lst.chunks(jsl, 5)) Fmt.print("$,12i", chunk)

System.print("\nFirst 20 Jacobsthal oblong numbers:") var oblongs = (0..19).map { |i| js[i] * js[i+1] }.toList for (chunk in Lst.chunks(oblongs, 5)) Fmt.print("$,14i", chunk)

var primes = js.where { |j| j.isProbablePrime(10) }.toList var count = primes.count var i = 31 while (count < 20) {

   var j = jacobsthal.call(i)
   if (j.isProbablePrime(10)) {
       primes.add(j)
       count = count + 1
   }
   i = i + 1

} System.print("\nFirst 20 Jacobsthal primes:") for (i in 0..19) Fmt.print("$i", primes[i])</lang>

Output:

First 30 Jacobsthal numbers:
           0            1            1            3            5
          11           21           43           85          171
         341          683        1,365        2,731        5,461
      10,923       21,845       43,691       87,381      174,763
     349,525      699,051    1,398,101    2,796,203    5,592,405
  11,184,811   22,369,621   44,739,243   89,478,485  178,956,971

First 30 Jacobsthal-Lucas numbers:
           2            1            5            7           17
          31           65          127          257          511
       1,025        2,047        4,097        8,191       16,385
      32,767       65,537      131,071      262,145      524,287
   1,048,577    2,097,151    4,194,305    8,388,607   16,777,217
  33,554,431   67,108,865  134,217,727  268,435,457  536,870,911

First 20 Jacobsthal oblong numbers:
             0              1              3             15             55
           231            903          3,655         14,535         58,311
       232,903        932,295      3,727,815     14,913,991     59,650,503
   238,612,935    954,429,895  3,817,763,271 15,270,965,703 61,084,037,575

First 20 Jacobsthal primes:
3
5
11
43
683
2731
43691
174763
2796203
715827883
2932031007403
768614336404564651
201487636602438195784363
845100400152152934331135470251
56713727820156410577229101238628035243
62357403192785191176690552862561408838653121833643
1046183622564446793972631570534611069350392574077339085483
267823007376498379256993682056860433753700498963798805883563
5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443

XPL0

<lang XPL0>func IsPrime(N); \Return 'true' if N is prime int N, I; [if N <= 2 then return N = 2; if (N&1) = 0 then \even >2\ return false; for I:= 3 to sqrt(N) do

   [if rem(N/I) = 0 then return false;
   I:= I+1;
   ];

return true; ];

proc Jaco(J2); \Display 30 Jacobsthal (or -Lucas) numbers real J2, J1, J; int N; [RlOut(0, J2); J1:= 1.0; RlOut(0, J1); for N:= 2 to 30-1 do

       [J:= J1 + 2.0*J2;
       RlOut(0, J);
       if rem((N+1)/5) = 0 then CrLf(0);
       J2:= J1;  J1:= J;
       ];

CrLf(0); ];

real J, J1, J2, JO; int N; [Format(14, 0); Jaco(0.0); Jaco(2.0); J2:= 1.0; RlOut(0, 0.0); J1:= 1.0; RlOut(0, J1); for N:= 2 to 20-1 do

       [J:= (J1 + 2.0*J2);
       JO:= J*J1;
       RlOut(0, JO);
       if rem((N+1)/5) = 0 then CrLf(0);
       J2:= J1;  J1:= J;
       ];

CrLf(0); J2:= 0.0; J1:= 1.0; N:= 0; loop [J:= J1 + 2.0*J2;

       if IsPrime(fix(J)) then
           [RlOut(0, J);
           N:= N+1;
           if rem(N/5) = 0 then CrLf(0);
           if N >= 10 then quit;
           ];
       J2:= J1;  J1:= J;
       ];

]</lang>

Output:

             0             1             1             3             5
            11            21            43            85           171
           341           683          1365          2731          5461
         10923         21845         43691         87381        174763
        349525        699051       1398101       2796203       5592405
      11184811      22369621      44739243      89478485     178956971

             2             1             5             7            17
            31            65           127           257           511
          1025          2047          4097          8191         16385
         32767         65537        131071        262145        524287
       1048577       2097151       4194305       8388607      16777217
      33554431      67108865     134217727     268435457     536870911

             0             1             3            15            55
           231           903          3655         14535         58311
        232903        932295       3727815      14913991      59650503
     238612935     954429895    3817763271   15270965703   61084037575

             3             5            11            43           683
          2731         43691        174763       2796203     715827883