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A100305 Expansion of (1 - x - 4*x^2)/(1 - 2*x - 7*x^2 + 8*x^3).

%I A100305 #18 Feb 04 2023 10:20:04
%S A100305 1,1,5,9,45,113,469,1369,5117,16065,56997,185513,641485,2125585,
%T A100305 7257461,24262137,82321821,276418913,934993477,3146344777,10626292589,
%U A100305 35797050801,120807391509,407183797913,1373642929981,4631113313281,15620256753125,52669163259369,177631217284365
%N A100305 Expansion of (1 - x - 4*x^2)/(1 - 2*x - 7*x^2 + 8*x^3).
%C A100305 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.
%H A100305 G. C. Greubel, <a href="/A100305/b100305.txt">Table of n, a(n) for n = 0..1000</a>
%H A100305 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,7,-8).
%F A100305 a(n) = 2*a(n-1) + 7*a(n-2) - 8*a(n-3).
%F A100305 a(n) = 1/2 + 2^(-n)*(sqrt(33)/132)*((1 + sqrt(33))^(n+1) - (1 - sqrt(33))^(n+1)).
%F A100305 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
%F A100305 a(n) = (1/2)*(1 + (2*sqrt(2)*i)^n*ChebyshevU(n, -i/(4*sqrt(2)))). - _G. C. Greubel_, Feb 03 2023
%t A100305 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 *)
%o A100305 (Magma) I:=[1,1,5]; [n le 3 select I[n] else 2*Self(n-1) +7*Self(n-2) -8*Self(n-3): n in [1..41]]; // _G. C. Greubel_, Feb 03 2023
%o A100305 (SageMath)
%o A100305 def A100305(n): return (1/2)*(1 + lucas_number1(n+1, 1, -8))
%o A100305 [A100305(n) for n in range(41)] # _G. C. Greubel_, Feb 03 2023
%Y A100305 Cf. A015443, A100304.
%Y A100305 Partial sums of A100303.
%K A100305 easy,nonn
%O A100305 0,3
%A A100305 _Paul Barry_, Nov 12 2004