# Jacobsthal number

In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence ${\displaystyle U_{n}(P,Q)}$ for which P = 1, and Q = −2[1]—and are defined by a similar recurrence relation: in simple terms, the sequence starts with 0 and 1, then each following number is found by adding the number before it to twice the number before that. The first Jacobsthal numbers are:

0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, … (sequence A001045 in the OEIS)

## Jacobsthal numbers

Jacobsthal numbers are defined by the recurrence relation:

${\displaystyle J_{n}={\begin{cases}0&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\J_{n-1}+2J_{n-2}&{\mbox{if }}n>1.\\\end{cases}}}$

The next Jacobsthal number is also given by the recursion formula:

${\displaystyle J_{n+1}=2J_{n}+(-1)^{n}\,,}$

or by:

${\displaystyle J_{n+1}=2^{n}-J_{n}.\,}$

The first recursion formula above is also satisfied by the powers of 2.

The Jacobsthal number at a specific point in the sequence may be calculated directly using the closed-form equation:[2]

${\displaystyle J_{n}={\frac {2^{n}-(-1)^{n}}{3}}.}$

The generating function for the Jacobsthal numbers is

${\displaystyle {\frac {x}{(1+x)(1-2x)}}.}$

The sum of the reciprocals of the Jacobsthal numbers is approximately 2.7186, slightly larger than e.

${\displaystyle J_{-n}=(-1)^{n}*J_{n}/2^{n}}$ (see )
${\displaystyle 2^{n}*(J_{-n}+J_{n})=3*(J_{n}^{2}=\,}$${\displaystyle (n))}$

## Jacobsthal-Lucas numbers

Jacobsthal-Lucas numbers represent the complementary Lucas sequence ${\displaystyle V_{n}(1,-2)}$. They satisfy the same recurrence relation as Jacobsthal numbers but have different initial values:

${\displaystyle j_{n}={\begin{cases}2&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\j_{n-1}+2j_{n-2}&{\mbox{if }}n>1.\\\end{cases}}}$

The following Jacobsthal-Lucas number also satisfies:[3]

${\displaystyle j_{n+1}=2j_{n}-3(-1)^{n}.\,}$

The Jacobsthal-Lucas number at a specific point in the sequence may be calculated directly using the closed-form equation:[3]

${\displaystyle j_{n}=2^{n}+(-1)^{n}.\,}$

The first Jacobsthal-Lucas numbers are:

2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, … (sequence A014551 in the OEIS).

## References

1. ^ Weisstein, Eric W. "Jacobsthal Number". MathWorld.
2. ^ Sloane, N. J. A. (ed.). "Sequence A001045 (Jacobsthal sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. ^ a b Sloane, N. J. A. (ed.). "Sequence A014551 (Jacobsthal-Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.