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A094855 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 4, s(2n+1) = 5.

%I A094855 #28 Feb 13 2022 06:30:08
%S A094855 1,3,10,35,124,440,1560,5525,19551,69142,244419,863788,3052100,
%T A094855 10782928,38092457,134560491,475313762,1678930611,5930320300,
%U A094855 20946860064,73987208296,261331829501,923052962407,3260318517230,11515766271219
%N A094855 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 4, s(2n+1) = 5.
%C A094855 In general, a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n+1) counts (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = j, s(2n+1) = k.
%C A094855 Counts all paths of length (2*n+1), n >= 0, starting at the initial node on the path graph P_8, see the Maple program. - _Johannes W. Meijer_, May 29 2010
%H A094855 Michael De Vlieger, <a href="/A094855/b094855.txt">Table of n, a(n) for n = 0..1824</a>
%H A094855 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,10,-1).
%F A094855 a(n) = (2/9)*Sum_{r=1..8} sin(4*r*Pi/9)*sin(5*r*Pi/9)*(2*cos(r*Pi/9))^(2n+1).
%F A094855 a(n) = 7*a(n-1) - 15*a(n-2) + 10*a(n-3) - a(n-4).
%F A094855 G.f.: (2*x-1)^2 / ( (x-1)*(x^3-9*x^2+6*x-1) ).
%F A094855 a(n) = A061551(2*n+1). - _Johannes W. Meijer_, May 29 2010
%p A094855 with(GraphTheory): G:=PathGraph(8): A:= AdjacencyMatrix(G): nmax:=24; for n from 0 to 2*nmax+2 do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..8); od: seq(a(2*n+1),n=0..nmax); # _Johannes W. Meijer_, May 29 2010
%t A094855 LinearRecurrence[{7,-15,10,-1},{1,3,10,35},30] (* _Harvey P. Dale_, Jan 17 2022 *)
%Y A094855 Odd bisection of A061551.
%Y A094855 Cf. A094854.
%K A094855 nonn,easy
%O A094855 0,2
%A A094855 _Herbert Kociemba_, Jun 13 2004