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—Jacobsthal numbers are the type 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, …

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:

$J\_\{n+1\}\; =\; 2J\_n\; +\; (-1)^n.\; ,$

The first recursion formula above is also satisfied by the powers of 2; the second is not.

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.

Jacobsthal-Lucas numbers

Jacobsthal-Lucas numbers retain the recurrence relation, L_n-1 + L_n-2, of Jacobsthal numbers, but use the starting conditions of the Lucas numbers, i.e. L₀ = 2, and L₁ = 1; they are defined by the recurrence relation:

$$

 L_n =

begin{cases} 2 & mbox{if } n = 0; 1 & mbox{if } n = 1; L_{n-1} + 2L_{n-2} & mbox{if } n > 1. end{cases}

The following Jacobsthal-Lucas number also satisfies:

$$

L_{n+1} = 2L_n - 3(-1)^n. ,

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

$$

 L_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, …

References