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—Jacobsthal numbers are the type for which "P" = 1, and "Q" = −2cite web
url = http://mathworld.wolfram.com/JacobsthalNumber.html
title = Jacobsthal Number
last = Weisstein
first = Eric W.
publisher = Wolfram Mathworld
date = 2006-05-15
accessdate = 2007-10-03] —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 OEIS|id=A001045 are:

:0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, …

Jacobsthal numbers

Jacobsthal numbers are defined by the recurrence relation:

:

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

: $J\_\{n+1\}\; =\; 2J\_n\; +\; (-1)^n.\; ,$

The first recursion formula above is also satisfied by the powers of 2; the second is not.

The Jacobsthal number at a specific point in the sequence may be calculated directly using the closed-form equation: [cite web
url = http://www.research.att.com/~njas/sequences/A001045
title = Jacobsthal sequence
last = Sloane
first = Neil J.A.
publisher = The On-Line Encyclopedia of Integer Sequences
date = 2007-10-01
accessdate = 2007-10-03]

:$J\_n\; =\; frac\{2^n\; -\; (-1)^n\}\; 3.$

Jacobsthal-Lucas numbers

Jacobsthal-Lucas numbers retain the recurrence relation, "L_n-1" + "L_n-2", of Jacobsthal numbers, but use the starting conditions of the Lucas numbers, i.e. "L₀" = 2, and "L₁" = 1; they are defined by the recurrence relation:

:

The following Jacobsthal-Lucas number also satisfies:cite web
url = http://www.research.att.com/~njas/sequences/A014551
title = Jacobsthal-Lucas numbers
last = Sloane
first = Neil J.A.
publisher = The On-Line Encyclopedia of Integer Sequences
date = 2007-10-03
accessdate = 2007-10-05]

:$L\_\{n+1\}\; =\; 2L\_n\; -\; 3(-1)^n.\; ,$

The Jacobsthal-Lucas number at a specific point in the sequence may be calculated directly using the closed-form equation:

:$L\_n\; =\; 2^n\; +\; (-1)^n.\; ,$

The first Jacobsthal-Lucas numbers OEIS|id=A014551 are:

:2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, 2047, 4097, 8191, 16385, 32767, 65537, 131071, 262145, 524287, 1048577, …

References