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%I #19 Apr 26 2023 08:54:53
%S 1,4,15,54,190,660,2276,7816,26776,91600,313104,1069728,3653728,
%T 12477504,42606656,145479808,496722304,1695962368,5790470400,
%U 19770087936,67499673088,230459040768,786837865472,2686435477504
%N 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) = 5.
%C 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 Michael De Vlieger, <a href="/A094821/b094821.txt">Table of n, a(n) for n = 1..1875</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,4).
%F a(n) = (1/4)*Sum_{r=1..7} sin(3*r*Pi/8)*sin(5*r*Pi/8)*(2*cos(r*Pi/8))^(2n).
%F a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3), n >= 4.
%F G.f.: -x*(x-1)^2 / ( (2*x-1)*(2*x^2-4*x+1) ).
%F 4*a(n) = 2*A007052(n) - 2^n. - _R. J. Mathar_, Nov 14 2019
%F E.g.f.: (exp(2*x)*(2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) - 1) - 1)/4. - _Stefano Spezia_, Apr 25 2023
%t Rest@ CoefficientList[Series[-x (x - 1)^2/((2 x - 1) (2 x^2 - 4 x + 1)), {x, 0, 24}], x] (* _Michael De Vlieger_, Feb 12 2022 *)
%Y Cf. A007052.
%K nonn,easy
%O 1,2
%A _Herbert Kociemba_, Jun 12 2004
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