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A100303 Expansion of (1 - x - 4*x^2)/(1 - x - 8*x^2).

%I A100303 #21 Aug 30 2024 21:09:52
%S A100303 1,0,4,4,36,68,356,900,3748,10948,40932,128516,455972,1484100,5131876,
%T A100303 17004676,58059684,194097092,658574564,2211351300,7479947812,
%U A100303 25170758212,85010340708,286376406404,966459132068,3257470383300,10989143439844,37048906506244,124962054024996
%N A100303 Expansion of (1 - x - 4*x^2)/(1 - x - 8*x^2).
%C A100303 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. (Spectrum : [0; 1^3; (1-sqrt(33))/2;(1+sqrt(33))/2]). a(n) counts closed walks of length n at each of the internal nodes.
%H A100303 G. C. Greubel, <a href="/A100303/b100303.txt">Table of n, a(n) for n = 0..1000</a>
%H A100303 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,8).
%F A100303 a(n) = 0^n/2 + (4/sqrt(33))*( ((1 + sqrt(33))/2)^(n-1) - ((1 - sqrt(33))/2)^(n-1) ).
%F A100303 a(n) = (1/2)*(A015443(n) - A015443(n-1)), n > 0. - _Ralf Stephan_, Jul 21 2013
%F A100303 E.g.f.: (33 + exp(x/2)*(33*cosh(sqrt(33)*x/2) - sqrt(33)*sinh(sqrt(33)*x/2)))/66. - _Stefano Spezia_, Sep 08 2022
%F A100303 a(n) = (1/2)*[n=0] + 4*A015443(n). - _G. C. Greubel_, Feb 04 2023
%t A100303 CoefficientList[Series[(1-x-4x^2)/(1-x-8x^2),{x,0,30}],x] (* _Harvey P. Dale_, Dec 01 2013 *)
%t A100303 LinearRecurrence[{1,8}, {1,0,4}, 31] (* _G. C. Greubel_, Feb 04 2023 *)
%o A100303 (Magma) [1] cat [n le 2 select 4*(n-1) else Self(n-1) +8*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Feb 04 2023
%o A100303 (SageMath)
%o A100303 def A100303(n): return (1/2)*int(n==0) + 4*lucas_number1(n-1, 1, -8)
%o A100303 [A100303(n) for n in range(31)] # _G. C. Greubel_, Feb 04 2023
%Y A100303 Cf. A015443, A100302, A100304.
%K A100303 easy,nonn
%O A100303 0,3
%A A100303 _Paul Barry_, Nov 12 2004