%I #19 Feb 12 2022 17:51:20
%S 1,4,14,48,165,571,1988,6953,24396,85786,302104,1064945,3756519,
%T 13256712,46796545,165225380,583440086,2060408640,7276716445,
%U 25700060995,90770326604,320598127113,1132355884236,3999522488002
%N Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 9 and s(i)  s(i1) = 1 for i = 1,2,...,2n+1, s(0) = 1, s(2n+1) = 4.
%C In general, a(n) = (2/m)*Sum_{r=1..m1} 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(i1) = 1 for i = 1,2,...,2n+1, s(0) = j, s(2n+1) = k.
%H Michael De Vlieger, <a href="/A094827/b094827.txt">Table of n, a(n) for n = 1..1825</a>
%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square ZigZag 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(4*r*Pi/9)*(2*cos(r*Pi/9))^(2*n+1).
%F a(n) = 7*a(n1)  15*a(n2) + 10*a(n3)  a(n4).
%F G.f.: x*(13*x+x^2) / ( (x1)*(x^39*x^2+6*x1) ).
%F 3*a(n) = A094829(n+2) 2*A094829(n+1) 2*A094829(n)1.  _R. J. Mathar_, Nov 14 2019
%t LinearRecurrence[{7,15,10,1},{1,4,14,48},30] (* _Harvey P. Dale_, Jul 09 2020 *)
%K nonn,easy
%O 1,2
%A _Herbert Kociemba_, Jun 13 2004
