JacobsthalSequence 2008-06-24 19:06:40 2008-06-29 18:02:20 Definition Jacobsthal number % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The {\em Jacobsthal sequence} is an additive sequence similar to the Fibonacci sequence, defined by the recurrence relation $J_n = J_{n - 1} + 2J_{n - 2}$, with initial terms $J_0 = 0$ and $J_1 = 1$. A number in the sequence is called a {\em 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 $n$th Jacobsthal number is the numerator of the alternating sum $$\sum_{i = 1}^n (-1)^{i - 1} \frac{1}{2^i}$$ (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, $$J_{2n} = \frac{2^{2n} - 1}{3}$$ and the other one for the odd-indexed terms, $$J_{2n + 1} = \frac{2^{2n + 1} - 2}{3} + 1.$$ These equations can be further generalized to $$J_n = \frac{(-1)^{n - 1} + 2^n}{3}.$$ The Jacobsthal numbers are named after the German mathematician Ernst Jacobsthal.