

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
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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;(1sqrt(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(n1) + 7*a(n2)  8*a(n3).
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[(1x4x^2)/(12x7x^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



