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%I #19 Feb 05 2023 02:00:13
%S 1,0,2,2,18,34,178,450,1874,5474,20466,64258,227986,742050,2565938,
%T 8502338,29029842,97048546,329287282,1105675650,3739973906,
%U 12585379106,42505170354,143188203202,483229566034,1628735191650,5494571719922,18524453253122,62481027012498
%N Expansion of (1 - x - 6*x^2)/(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 '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 extremity nodes.
%H G. C. Greubel, <a href="/A100304/b100304.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,8).
%F a(n) = 3*0^n/4 + (2/sqrt(33))*( ((1 + sqrt(33))/2)^(n-1) - ((1 - sqrt(33))/2)^(n-1) ).
%F E.g.f.: (99 + 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) = (3/4)*[n=0] + 2*A015443(n-2). - _G. C. Greubel_, Feb 04 2023
%t LinearRecurrence[{1,8},{1,0,2},27] (* _Stefano Spezia_, Sep 08 2022 *)
%o (Magma) [1] cat [n le 2 select 2*(n-1) else Self(n-1) +8*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Feb 04 2023
%o (SageMath)
%o def A100304(n): return (3/4)*int(n==0) + 2*lucas_number1(n-1, 1, -8)
%o [A100304(n) for n in range(31)] # _G. C. Greubel_, Feb 04 2023
%Y Essentially half A100303.
%Y Cf. A015443, A100302 (partial sums), A100305.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Nov 12 2004
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