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%I #22 Feb 12 2022 17:51:37
%S 1,5,20,75,274,988,3536,12597,44745,158632,561683,1987154,7026408,
%T 24835744,87763945,310088381,1095490524,3869911659,13670143618,
%U 48287147300,170561502896,602454835293,2127962632993,7516243783216
%N Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, 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))^(2*n) 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="/A094828/b094828.txt">Table of n, a(n) for n = 2..1826</a>
%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-15,10,-1).
%F a(n) = (2/9)*Sum_{r=1..8} sin(r*Pi/9)*sin(5*r*Pi/9)*(2*cos(r*Pi/9))^(2*n).
%F a(n) = 7*a(n-1) - 15*a(n-2) + 10*a(n-3) - a(n-4).
%F G.f.: -x^2*(-1+2*x) / ( (x-1)*(x^3-9*x^2+6*x-1) ).
%F a(n) = A094256(n-1) - 2*A094256(n-2). - _R. J. Mathar_, Nov 14 2019
%F 3*a(n) = A094829(n+2) -5*A094829(n+1)+7*A094829(n)-1. - _R. J. Mathar_, Nov 14 2019
%t LinearRecurrence[{7,-15,10,-1},{1,5,20,75},30] (* _Harvey P. Dale_, Apr 27 2020 *)
%K nonn,easy
%O 2,2
%A _Herbert Kociemba_, Jun 13 2004
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