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)

A Jacobsthal prime is a Jacobsthal number that is also prime. The first Jacobsthal primes are:

3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, … (sequence A049883 in the OEIS)

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 second 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.

The Jacobsthal numbers can be extended to negative indices using the recurrence relation or the explicit formula, giving

${\displaystyle J_{-n}=(-1)^{n+1}J_{n}/2^{n}}$ (see OEIS: A077925)

The following identities holds

${\displaystyle 2^{n}(J_{-n}+J_{n})=3J_{n}^{2}}$ (see OEIS: A139818)

${\displaystyle J_{n}=F_{n}+\sum _{i=0}^{n-2}J_{i}F_{n-i-1}}$ where ${\displaystyle F_{n}}$ is the nth Fibonacci number.

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:^[2]

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

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

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

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).

The first Jacobsthal Oblong numbers are: 0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, … (sequence A084175 in the OEIS)

${\displaystyle Jo_{n}=J_{n}J_{n+1}}$