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%I #18 Jan 30 2024 18:30:56
%S 1,3,12,60,336,1968,11712,70080,420096,2519808,15117312,90700800,
%T 544198656,3265179648,19591053312,117546270720,705277526016,
%U 4231664959488,25389989363712,152339935395840,914039610802176
%N Expansion of (1-5*x)/((1-2*x)*(1-6*x)).
%C Second binomial transform of expansion of (1-3*x)/(1-4*x). Third binomial transform of A054878 (closed walks at a vertex of K_4). With interpolated zeros, counts closed walks of length n at the vertices of the edge-vertex incidence graph of K_4 associated with the vertices of K_4.
%H G. C. Greubel, <a href="/A092803/b092803.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 (8,-12).
%F a(n) = 2^(n-2)*(3^n + 3) = (6^n + 3*2^n)/4.
%F G.f.: U(0)/4 where U(k)= 1 + 2/(3^k - 3^k/(2 + 1 - 12*x*3^k/(6*x*3^k + 1/U(k+1)))) ; (continued fraction, 4-step). - _Sergei N. Gladkovskii_, Oct 30 2012
%F G.f.: U(0)/4 where U(k)= 1 + 3/( 3^k - 2*x*9^k/(2*x*3^k + 1/U(k+1))) ; (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Oct 31 2012
%F E.g.f.: (1/4)*( 3*exp(2*x) + exp(6*x) ). - _G. C. Greubel_, Jan 04 2023
%t LinearRecurrence[{8,-12}, {1,3}, 41] (* _G. C. Greubel_, Jan 04 2023 *)
%t CoefficientList[Series[(1-5x)/((1-2x)(1-6x)),{x,0,30}],x] (* _Harvey P. Dale_, Jan 30 2024 *)
%o (Magma) [3*(6^(n-1) + 2^(n-1))/2: n in [0..40]]; // _G. C. Greubel_, Jan 04 2023
%o (SageMath) [3*(6^(n-1) +2^(n-1))/2 for n in range(41)] # _G. C. Greubel_, Jan 04 2023
%Y Cf. A054878, A092807.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 06 2004
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