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A100305 Expansion of (1 - x - 4*x^2)/(1 - 2*x - 7*x^2 + 8*x^3). 1
1, 1, 5, 9, 45, 113, 469, 1369, 5117, 16065, 56997, 185513, 641485, 2125585, 7257461, 24262137, 82321821, 276418913, 934993477, 3146344777, 10626292589, 35797050801, 120807391509, 407183797913, 1373642929981, 4631113313281, 15620256753125, 52669163259369, 177631217284365 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Construct a graph as follows: form the graph whose adjacency matrix is the tensor product of that of P_3 and [1,1;1,1], then add a loop at each of the 'internal' nodes. (Spectrum : [0^3;1;(1-sqrt(33))/2;(1+sqrt(33))/2]). a(n) counts closed walks of length n at each of the 'internal' nodes.

LINKS

Table of n, a(n) for n=0..28.

Index entries for linear recurrences with constant coefficients, signature (2,7,-8).

FORMULA

a(n) = 2*a(n-1) + 7*a(n-2) - 8*a(n-3).

a(n) = 1/2 + ((sqrt(33) + 1)^(n+1) + (sqrt(33) - 1)^(n+1)*(-1)^n)*sqrt(33)*2^(-n)/132.

E.g.f.: exp(x)/2 + exp(x/2)*(33*cosh(sqrt(33)*x/2) + sqrt(33)*sinh(sqrt(33)*x/2))/66. - Stefano Spezia, Sep 08 2022

MATHEMATICA

CoefficientList[Series[(1-x-4x^2)/(1-2x-7x^2+8x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 7, -8}, {1, 1, 5}, 40] (* Harvey P. Dale, Oct 05 2012 *)

CROSSREFS

Cf. A100304.

Partial sums of A100303.

Sequence in context: A149497 A149498 A149499 * A149500 A149501 A149502

Adjacent sequences: A100302 A100303 A100304 * A100306 A100307 A100308

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 12 2004

STATUS

approved

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Last modified November 30 12:40 EST 2022. Contains 358441 sequences. (Running on oeis4.)