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—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:

Jacobsthal numbers are defined by the recurrence relation:

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

or by:

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:

The generating function for the Jacobsthal numbers is

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:

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