# Jacobsthal numbers

Jacobsthal numbers
You are encouraged to solve this task according to the task description, using any language you may know.

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 Jn by its direct successor Jn+1.

Jacobsthal primes are Jacobsthal numbers that are prime.

• 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

## ALGOL 68

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

<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

<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
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

```   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)
```

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>
```

## 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);
```

}

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)
}
```

}

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>

<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] + .;

1. If \$j is 0, then an error condition is raised;
2. otherwise, assuming infinite-precision integer arithmetic,
3. 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] ;
```
1. To take advantage of gojq's arbitrary-precision integer arithmetic:

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

``` . 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 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))
```

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_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```

## 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
```

## 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)) {
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
```