Jacobsthal numbers

Jacobsthal numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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
   Jn = Jn-1 + 2 × Jn-2

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

   Jn = ( 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:

   
   JLn = 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

C

Library: GMP

<lang c>#include <stdio.h>

1. 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

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

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>

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)

lazyprintnum(s) = println(replace("$s", r"[^\d\,\s]" => ""))

lazyprintnum(collect(take(30, Jacobsthal))) lazyprintnum(collect(take(30, JLucas))) lazyprintnum(collect(take(20, Joblong))) lazyprintnum(collect(take(15, Jprimes)))

</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, 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>

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

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