Jacobsthal number

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:

: J_n = egin{cases} 0 & mbox{if } n = 0; \ 1 & mbox{if } n = 1; \ J_{n-1} + 2J_{n-2} & mbox{if } n > 1. \ end{cases}

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, "Ln-1" + "Ln-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:

: L_n = egin{cases} 2 & mbox{if } n = 0; \ 1 & mbox{if } n = 1; \ L_{n-1} + 2L_{n-2} & mbox{if } n > 1. \ end{cases}

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


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Número de Jacobsthal — En matemáticas, los números de Jacobsthal son una sucesión de números enteros nombrada en honor al matemático alemán Ernst Jacobsthal. Esta sucesión tiene relación con la de Fibonacci, de hecho, es un caso particular de sucesión de Lucas en el… …   Wikipedia Español

  • Ernst Jacobsthal — Ernst Erich Jacobsthal (16 October 1882, Berlin – 6 February 1965, Überlingen)cite web url = http://www.numbertheory.org/obituaries/OTHERS/jacobsthal eng.html title = Ernst Jacobsthal last = Selberg first = Sigmund language = English, translated… …   Wikipedia

  • Suite de Lucas — En mathématiques, une suite de Lucas est une généralisation de la suite de Fibonacci et des nombres de Lucas. Les suites de Lucas furent étudiées en premier par le mathématicien français Édouard Lucas. Sommaire 1 Relations de récurrence 2 Terme… …   Wikipédia en Français

  • Lucas sequence — In mathematics, a Lucas sequence is a particular generalisation of the Fibonacci numbers and Lucas numbers. Lucas sequences are named after French mathematician Edouard Lucas. Recurrence relations Given two integer parameters P and Q which… …   Wikipedia

  • Последовательность Люка — Не следует путать с числами Люка. В математике, последовательностями Люка называют семейство пар линейных рекуррентных последовательностей второго порядка, впервые рассмотренных Эдуардом Люка. Последовательности Люка представляют собой пары… …   Википедия

  • List of mathematics articles (J) — NOTOC J J homomorphism J integral J invariant J. H. Wilkinson Prize for Numerical Software Jaccard index Jack function Jacket matrix Jackson integral Jackson network Jackson s dimensional theorem Jackson s inequality Jackson s theorem Jackson s… …   Wikipedia

  • Paley construction — In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley.The Paley construction uses quadratic residues in a… …   Wikipedia

  • Celts — Celt redirects here. For other uses, see Celt (disambiguation). This article is about the ancient peoples of Europe. For Celts of the present day, see Celts (modern). Diachronic distribution of Celtic peoples …   Wikipedia

  • Liste der Staatsstraßen in Sachsen — Stationszeichen in Sachsen Ausschild …   Deutsch Wikipedia

  • painting, Western — ▪ art Introduction       history of Western painting from its beginnings in prehistoric times to the present.       Painting, the execution of forms and shapes on a surface by means of pigment (but see also drawing for discussion of depictions in …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”