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%I #19 Mar 11 2020 17:18:22
%S 0,0,0,0,0,6,8,28,44,100,162,318,514,942,1518,2672,4302,7380,11882,
%T 20040,32276,53810,86710,143396,231204,380152,613286,1004188,1620864,
%U 2645928,4272744,6959326,11242518,18281222,29542078,47978666,77552928,125836374,203445784
%N From random walks on complete directed triangle.
%H Sean A. Irvine, <a href="/A007829/b007829.txt">Table of n, a(n) for n = 0..500</a>
%H E. Bussian, <a href="/A007829/a007829.pdf">Email to N. J. A. Sloane, Oct. 1994</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-6,-4,5,5,-2,-2).
%F From _Colin Barker_, Feb 03 2018: (Start)
%F G.f.: 2*x^5*(3 - 2*x - 3*x^2) / ((1 - x)*(1 - x - x^2)*(1 - 2*x^2)*(1 - x^2 - x^3)).
%F a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3) - 4*a(n-4) + 5*a(n-5) + 5*a(n-6) - 2*a(n-7) - 2*a(n-8) for n>7.
%F (End)
%F From _G. C. Greubel_, Mar 11 2020: (Start)
%F a(n) = 2*(2 + Fibonacci(n+2) - 2^floor(n/2) - A084338(n+2)).
%F a(n) = 2*(2 + Fibonacci(n+2) - 2^floor(n/2) - b(n+7) - b(n+5)), where b(n) = A000931(n). (End)
%p m:=35; S:=series(2*x^5*(3-2*x-3*x^2)/((1-x)*(1-x-x^2)*(1-2*x^2)*(1-x^2-x^3)), x, m+1): seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Mar 11 2020
%t b[n_]:= b[n]= If[n==0, 1, If[n<3, 0, b[n-2] +b[n-3]]]; Table[2*(2 +Fibonacci[n+2] -2^Floor[n/2] -p[n+7] -p[n+5]), {n,0,35}] (* _G. C. Greubel_, Mar 11 2020 *)
%o (Sage)
%o def A007829_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 2*x^5*(3-2*x-3*x^2)/((1-x)*(1-x-x^2)*(1-2*x^2)*(1-x^2-x^3)) ).list()
%o A007829_list(35) # _G. C. Greubel_, Mar 11 2020
%Y Cf. A000931, A084338.
%K nonn,walk
%O 0,6
%A Eric Bussian [ ebussian(AT)math.gatech.edu ]
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