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A113405 Expansion of x^3/(1-2*x+x^3-2*x^4) = x^3/( (1-2*x)*(1+x)*(1-x+x^2) ). 23
0, 0, 0, 1, 2, 4, 7, 14, 28, 57, 114, 228, 455, 910, 1820, 3641, 7282, 14564, 29127, 58254, 116508, 233017, 466034, 932068, 1864135, 3728270, 7456540, 14913081, 29826162, 59652324, 119304647, 238609294, 477218588, 954437177, 1908874354, 3817748708 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

A transform of the Jacobsthal numbers. A059633 is the equivalent transform of the Fibonacci numbers.

Paul Curtz, Aug 05 2007, observes that the inverse binomial transform of 0,0,0,1,2,4,7,14,28,57,114,228,455,910,1820,... gives the same sequence up to signs. That is, the extended sequence is an eigen-sequence for the inverse binomial transform (an autosequence).

The round() function enables the closed (non-recurrence) formula to take a very simple form: see Formula section. This can be generalized without loss of simplicity to a(n) = round(b^n/c), where b and c are very small, incommensurate integers (c may also be an integer fraction). Particular choices of small integers for b and c produce a number of well-known sequences which are usually defined by a recurrence - see Cross Reference. - Ross Drewe, Sep 03 2009

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, (arXiv:math/0205301 [math.CO], 2002) [Link to arXiv version]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

N. J. A. Sloane, Transforms

FORMULA

a(n) = 2a(n-1)-a(n-3)+2a(n-4).

a(n) = sum{k=0..floor(n/2), C(n-k,k) A001045(k)}.

a(n) = sum{k=0..n, C((n+k)/2,k) A001045((n-k)/2)(1+(-1)^(n-k))/2}.

a(3n) = A015565(n), a(3n+1) = 2*A015565(n), a(3n+2) = 4*A015565(n). - Paul Curtz, Nov 30 2007

a(n+1)-2a(n)= A131531(n). a(n)+a(n+3)=2^n. - Paul Curtz, Dec 16 2007

a(n) = round(2^n/9). - Ross Drewe, Sep 03 2009

MAPLE

A010892 := proc(n) op((n mod 6)+1, [1, 1, 0, -1, -1, 0]) ; end proc:

A113405 := proc(n) (2^n-(-1)^n)/9 -A010892(n-1)/3; end proc: # R. J. Mathar, Dec 17 2010

MATHEMATICA

CoefficientList[Series[x^3/(1-2x+x^3-2x^4), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 0, -1, 2}, {0, 0, 0, 1}, 40] (* Harvey P. Dale, Apr 30 2011 *)

PROG

(PARI) a(n)=2^n\/9 \\ Charles R Greathouse IV, Jun 05, 2011

(MAGMA) [Round(2^n/9): n in [0..40]]; // Vincenzo Librandi, Aug 11 2011

CROSSREFS

From Ross Drewe, Sep 03 2009: (Start)

Other sequences a(n) = round(b^n / c), where b and c are very small integers:

A001045 b = 2; c = 3

A007910 b = 2; c = 5

A016029 b = 2; c = 5/3

A077947 b = 2; c = 7

abs(A078043) b = 2; c = 7/3

A007051 b = 3; c = 2

A015518 b = 3; c = 4

A034478 b = 5; c = 2

A003463 b = 5; c = 4

A015531 b = 5; c = 6

(End)

Sequence in context: A251744 A251751 A251765 * A119340 A119341 A119342

Adjacent sequences:  A113402 A113403 A113404 * A113406 A113407 A113408

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Oct 28 2005

EXTENSIONS

Edited by N. J. A. Sloane, Dec 13 2007

STATUS

approved

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Last modified October 22 02:06 EDT 2017. Contains 293756 sequences.