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A094806 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) = 1, s(2n) = 5.

%I A094806 #27 Feb 12 2022 17:58:20
%S A094806 1,5,20,74,264,924,3200,11016,37792,129392,442496,1512224,5165952,
%T A094806 17643456,60250112,205729920,702452224,2398414592,8188884992,
%U A094806 27958972928,95458646016,325917686784,1112755552256,3799191029760,12971261403136,44286680330240
%N A094806 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) = 1, s(2n) = 5.
%C A094806 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 A094806 Harvey P. Dale, <a href="/A094806/b094806.txt">Table of n, a(n) for n = 2..1000</a>
%H A094806 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 A094806 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,4).
%F A094806 a(n) = (1/4)*Sum_{k=1..7} sin(Pi*k/8)*sin(5*Pi*k/8)*(2*cos(Pi*k/8))^(2n).
%F A094806 a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3).
%F A094806 G.f.: x^2*(x-1) / ( (2*x-1)*(2*x^2-4*x+1) ).
%F A094806 a(n) = (-2^n+(-(2-sqrt(2))^n+(2+sqrt(2))^n)/sqrt(2))/4. - _Colin Barker_, Apr 27 2016
%F A094806 4*a(n) = 2*A007070(n-1) - 2^n.- _R. J. Mathar_, Nov 14 2019
%t A094806 f[n_] := FullSimplify[ TrigToExp[(1/4)Sum[ Sin[Pi*k/8]Sin[5Pi*k/8](2Cos[Pi*k/8])^(2n), {k, 1, 7}]]]; Table[ f[n], {n, 2, 25}] (* _Robert G. Wilson v_, Jun 18 2004 *)
%t A094806 LinearRecurrence[{6,-10,4},{1,5,20},30] (* _Harvey P. Dale_, Mar 04 2015 *)
%K A094806 nonn,easy
%O A094806 2,2
%A A094806 _Herbert Kociemba_, Jun 11 2004