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A094833 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) = 3, s(2n) = 5.

%I A094833 #18 Jul 02 2021 16:46:19
%S A094833 1,4,15,55,199,714,2548,9061,32148,113887,403051,1425471,5039254,
%T A094833 17809336,62928201,222324436,785402143,2774421135,9800231959,
%U A094833 34617003682,122274355596,431893332397,1525507797700,5388281150223
%N A094833 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) = 3, s(2n) = 5.
%C A094833 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 A094833 Michael De Vlieger, <a href="/A094833/b094833.txt">Table of n, a(n) for n = 1..1825</a>
%H A094833 Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H A094833 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,1).
%F A094833 a(n) = (2/9)*Sum_{r=1..8} sin(r*Pi/3)*sin(5*r*Pi/9)*(2*cos(r*Pi/9))^(2n).
%F A094833 a(n) = 6a(n-1) - 9a(n-2) + a(n-3).
%F A094833 G.f.: (-x+2x^2)/(-1 + 6x - 9x^2 + x^3).
%F A094833 a(n+1) = 3*a(n) + A094832(n-1). - _Philippe Deléham_, Mar 20 2007
%F A094833 a(n) = A094829(n+1) - 2*A094829(n). - _R. J. Mathar_, Nov 14 2019
%t A094833 Rest@ CoefficientList[Series[(-x + 2 x^2)/(-1 + 6 x - 9 x^2 + x^3), {x, 0, 24}], x] (* _Michael De Vlieger_, Jul 02 2021 *)
%K A094833 nonn,easy
%O A094833 1,2
%A A094833 _Herbert Kociemba_, Jun 13 2004