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%I #14 Feb 04 2023 19:02:40
%S 1,1,3,5,23,57,235,685,2559,8033,28499,92757,320743,1062793,3628731,
%T 12131069,41160911,138209457,467496739,1573172389,5313146295,
%U 17898525401,60403695755,203591898957,686821464991,2315556656641,7810128376563,26334581629685,88815608642183
%N Expansion of (1 - x - 6*x^2)/((1 - x)*(1 - x - 8*x^2)).
%C 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 extremity nodes. a(n) counts closed walks of length n at each of the extremity nodes.
%H G. C. Greubel, <a href="/A100302/b100302.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,7,-8).
%F a(n) = 2*a(n-1) + 7*a(n-2) - 8*a(n-3).
%F a(n) = 3/4 + (1/(4*sqrt(33))*( ((1 + sqrt(33))/2)^(n+1) - ((1 - sqrt(33))/2)^(n+1)).
%F E.g.f.: 3*exp(x)/4 + exp(x/2)*(33*cosh(sqrt(33)*x/2) + sqrt(33)*sinh(sqrt(33)*x/2))/132. - _Stefano Spezia_, Sep 08 2022
%F a(n) = (1/4)*(3 + A015443(n+1)). - _G. C. Greubel_, Feb 04 2023
%t LinearRecurrence[{2,7,-8},{1,1,3},29] (* _Stefano Spezia_, Sep 08 2022 *)
%o (Magma) I:=[1,1,3]; [n le 3 select I[n] else 2*Self(n-1) +7*Self(n-2) -8*Self(n-3): n in [1..31]]; // _G. C. Greubel_, Feb 04 2023
%o (SageMath)
%o def A100302(n): return (3 + lucas_number1(n+1, 1, -8))/4
%o [A100302(n) for n in range(31)] # _G. C. Greubel_, Feb 04 2023
%Y Cf. A015443, A100303, A100304.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Nov 12 2004
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