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A007829 From random walks on complete directed triangle.

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