# Jacobsthal sequence

The Jacobsthal sequence is an additive^{} sequence^{} similar to the Fibonacci sequence^{}, defined by the recurrence relation ${J}_{n}={J}_{n-1}+2{J}_{n-2}$, 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.

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 |