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A026597 Expansion of (1+x)/(1-x-4*x^2). 23
1, 2, 6, 14, 38, 94, 246, 622, 1606, 4094, 10518, 26894, 68966, 176542, 452406, 1158574, 2968198, 7602494, 19475286, 49885262, 127786406, 327327454, 838473078, 2147782894, 5501675206, 14092806782, 36099507606, 92470734734 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence can generated by the following formula: a(n) = a(n-1) + 4*a(n-2) when n > 2; a[1] = 1, a[2] = 2. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 21 2004

An elephant sequence, see A175654 and A175655. For the corner squares just one A[5] vector, with decimal value 325, leads to the sequence given above. For the central square this vector leads to a companion sequence that is 4 times this very same sequence with n >= -1. - Johannes W. Meijer, Aug 15 2010

Equals INVERTi transform of A180168. [Gary W. Adamson, Aug 14 2010]

Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 2 X 2 cells and remove the cells that have one '1' in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.357. - Peter Karpov, Apr 20 2017

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 0..500

Peter Karpov, InvMem, Item 26

Peter Karpov, Illustration of initial terms (n = 1..8)

Index entries for linear recurrences with constant coefficients, signature (1,4).

FORMULA

G.f.: (1+x)/(1-x-4*x^2).

a(n) = T(n,0) + T(n,1) + ... + T(n,2*n), T given by A026584.

a(n) = sum(k=0..n, binomial(floor((2*n-k-1)/2), n-k)*2^k ). - Paul Barry, Feb 11 2005

a(n) = A006131(n)+A006131(n-1), n>=1. - R. J. Mathar, Oct 20 2006

a(n) = sum(k=0..n, C(floor((2*n-k)/2),n-k)*4^floor(k/2) ). - Paul Barry, Feb 02 2007

Inverse binomial transform of A007482: (1, 3, 11, 39, 139, 495,...). - Gary W. Adamson, Dec 04 2007

a(n) = sum(0<=k<=n+1, A122950(n+1,k)*3^(n+1-k) ). - Philippe Deléham, Jan 04 2008

a(n) = (1/2+3*sqrt(17)/34)*(1/2+sqrt(17)/2)^n+(1/2-3*sqrt(17)/34)*(1/2-sqrt(17)/2)^n. - Antonio Alberto Olivares, Jun 07 2011

MATHEMATICA

LinearRecurrence[{1, 4}, {1, 2}, 40] (* Harvey P. Dale, Nov 28 2011 *)

CROSSREFS

Cf. A006131, A006138, A007482, A026581.

Cf. A180168. [Gary W. Adamson, Aug 14 2010]

Sequence in context: A263758 A100067 * A122112 A190788 A168259 A275208

Adjacent sequences:  A026594 A026595 A026596 * A026598 A026599 A026600

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

EXTENSIONS

Better name from Ralf Stephan, Jul 14 2013

STATUS

approved

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Last modified June 27 11:25 EDT 2017. Contains 288788 sequences.