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# Jacobsthal sequence

The 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 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.

## Mathematics Subject Classification

11B39*no label found*

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