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 $U_{n}(P,Q)$ for which P = 1, and Q = −2—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:

$J_{n}={\begin{cases}0&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\J_{n-1}+2J_{n-2}&{\mbox{if }}n>1.\\\end{cases}}$ The next Jacobsthal number is also given by the recursion formula:

$J_{n+1}=2J_{n}+(-1)^{n}\,,$ or by:

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

$J_{n}={\frac {2^{n}-(-1)^{n}}{3}}.$ The generating function for the Jacobsthal numbers is

${\frac {x}{(1+x)(1-2x)}}.$ The sum of the reciprocals of the Jacobsthal numbers is approximately 2.7186, slightly larger than e.

Jacobsthal-Lucas numbers

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

$j_{n}={\begin{cases}2&{\mbox{if }}n=0;\\1&{\mbox{if }}n=1;\\j_{n-1}+2j_{n-2}&{\mbox{if }}n>1.\\\end{cases}}$ The following Jacobsthal-Lucas number also satisfies:

$j_{n+1}=2j_{n}-3(-1)^{n}.\,$ The Jacobsthal-Lucas number at a specific point in the sequence may be calculated directly using the closed-form equation:

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