Jacobsthal sequence

The Jacobsthal sequence is an additivePlanetmathPlanetmath sequenceMathworldPlanetmath similar to the Fibonacci sequenceMathworldPlanetmath, defined by the recurrence relation Jn=Jn-1+2Jn-2, with initial terms J0=0 and J1=1. A number in the sequence is called a Jacobsthal number. The first few are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, etc., listed in A001045 of Sloane’s OEIS.

The nth Jacobsthal number is the numerator of the alternating sum


(the denominators are powers of two). This suggests a closed form: by putting the series solution over a common denominator and summing the geometric series in the numerator, we obtain two equations, one for even-indexed terms of the sequence,


and the other one for the odd-indexed terms,


These equations can be further generalized to


The Jacobsthal numbers are named after the German mathematician Ernst Jacobsthal.

Title Jacobsthal sequence
Canonical name JacobsthalSequence
Date of creation 2013-03-22 18:09:40
Last modified on 2013-03-22 18:09:40
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 11B39
Defines Jacobsthal number