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A094817 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.

%I A094817 #21 Feb 13 2022 02:37:20
%S A094817 2,6,19,62,206,692,2340,7944,27032,92112,314128,1071776,3657824,
%T A094817 12485696,42623040,145512576,496787840,1696093440,5790732544,
%U A094817 19770612224,67500721664,230461137920,786842059776,2686443866112
%N A094817 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3.
%C A094817 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) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.
%H A094817 Michael De Vlieger, <a href="/A094817/b094817.txt">Table of n, a(n) for n = 1..1875</a>
%H A094817 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,4).
%F A094817 a(n) = (1/4) * Sum_{r=1..7} sin(3*r*Pi/8)^2*(2*cos(r*Pi/8))^(2*n).
%F A094817 a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3), n >= 4.
%F A094817 G.f.: -x*(2-6*x+3*x^2) / ( (2*x-1)*(2*x^2-4*x+1) ).
%F A094817 a(n) = A216232(n,n), for n >= 1. - _Philippe Deléham_, Mar 21 2013
%F A094817 4*a(n) = 2*A007052(n) + 2^n. - _R. J. Mathar_, Nov 14 2019
%t A094817 Rest@ CoefficientList[Series[-x (2 - 6 x + 3 x^2)/((2 x - 1) (2 x^2 - 4 x + 1)), {x, 0, 24}], x] (* _Michael De Vlieger_, Feb 12 2022 *)
%Y A094817 Cf. A007052, A216232.
%K A094817 nonn,easy
%O A094817 1,1
%A A094817 _Herbert Kociemba_, Jun 12 2004