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%I #15 Oct 29 2019 12:17:05
%S 1,3,10,30,88,252,712,1992,5536,15312,42208,116064,318592,873408,
%T 2392192,6547584,17912320,48985344,133926400,366085632,1000548352,
%U 2734316544,7471826944,20416481280,55785005056,152419749888
%N Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 4.
%C In general, a(n,m,j,k) = (2/m)*Sum_{r=1..m-1} Sin(j*r*Pi/m)*Sin(k*r*Pi/m)*(1+2*cos(Pi*r/m))^n) is the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < m and |s(i) -s(i-1)| <= 1 for i = 1,2,...,n, s(0) = j, s(n) = k.
%H Colin Barker, <a href="/A094306/b094306.txt">Table of n, a(n) for n = 2..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4).
%F a(n) = ((1-sqrt(3))^n + (1+sqrt(3))^n - 2^n)/4.
%F a(n) = (1/3)*Sum_{k=1..5} Sin(Pi*k/3)*Sin(2*Pi*k/3)*(1+2*cos(Pi*k/6))^n.
%F From _Colin Barker_, Oct 29 2019: (Start)
%F G.f.: x^2*(1 - x) / ((1 - 2*x)*(1 - 2*x - 2*x^2)).
%F a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) for n>4.
%F (End)
%t f[n_] := FullSimplify[ TrigToExp[(1/3)Sum[ Sin[Pi*k/3] Sin[2Pi*k/3] (1 + 2Cos[Pi*k/6])^n, {k, 1, 5}]]]; Table[ f[n], {n, 2, 27}] (* _Robert G. Wilson v_, Jun 18 2004 *)
%o (PARI) Vec(x^2*(1 - x) / ((1 - 2*x)*(1 - 2*x - 2*x^2)) + O(x^35)) \\ _Colin Barker_, Oct 29 2019
%K easy,nonn
%O 2,2
%A _Herbert Kociemba_, Jun 02 2004