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%I #45 Jan 21 2023 11:47:20
%S 1,0,4,0,16,0,64,0,256,0,1024,0,4096,0,16384,0,65536,0,262144,0,
%T 1048576,0,4194304,0,16777216,0,67108864,0,268435456,0,1073741824,0,
%U 4294967296,0,17179869184,0,68719476736,0,274877906944,0
%N Number of round trips of length n on the cycle graph C_2 from any of the two vertices.
%C See the array and triangle A199571 for the general cycle graph C_N counting.
%C This is A000302 and A000004 interleaved. - _Omar E. Pol_, Nov 09 2011
%C With offset = 1: Number of ways to separate n distinguishable objects into an odd size pile and an even size pile. For example: a(3) = 4 because we have: {{1},{2,3}}; {{2},{1,3}}; {{3},{1,2}}; {{1,2,3},{}}. - _Geoffrey Critzer_, Jun 10 2013
%C Inverse Stirling transform of A065143. - _Vladimir Reshetnikov_, Nov 01 2015
%H R. J. Mathar, <a href="/A102518/a102518.pdf">Counting Walks on Finite Graphs</a>, Section 1.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,4).
%F a(n) = (2^n + (-2)^n)/2 = 2^(n-1)*(1 + (-1)^n)), n >= 0.
%F O.g.f.: 1/(1-(2*x)^2).
%F E.g.f.: cosh(2*x)=U(0) where U(k) = 1 + 2*x^2/((4*k+1)*(2*k+1) - x^2*(4*k+1)*(2*k+1)/(x^2 + (4*k+3)*(k+1)/U(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Oct 23 2012
%e a(2) = 4 from starting with vertex no. 1, with edges e1 and e2 to vertex no. 2: e1e1, e2e2, e1e2 and e2e1.
%t nn = 39; Drop[Range[0, nn]! CoefficientList[Series[ Sinh[x] Cosh[x], {x, 0, nn}],x], 1] (* _Geoffrey Critzer_, Jun 10 2013 *)
%o (PARI) vector(100, n, n--; (2^(n) +(-2)^n)/2) \\ _Altug Alkan_, Nov 02 2015
%Y Cf. A000007 (N=1), A078008 (N=3). a(n) is second row of array w(N,L) A199571, and second column of the triangle a(K,N) A199571.
%Y Cf. A065143 (Stirling transform).
%K nonn,easy,walk
%O 0,3
%A _Wolfdieter Lang_, Nov 08 2011
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