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A199572
Number of round trips of length n on the cycle graph C_2 from any of the two vertices.
8
1, 0, 4, 0, 16, 0, 64, 0, 256, 0, 1024, 0, 4096, 0, 16384, 0, 65536, 0, 262144, 0, 1048576, 0, 4194304, 0, 16777216, 0, 67108864, 0, 268435456, 0, 1073741824, 0, 4294967296, 0, 17179869184, 0, 68719476736, 0, 274877906944, 0
OFFSET
0,3
COMMENTS
See the array and triangle A199571 for the general cycle graph C_N counting.
This is A000302 and A000004 interleaved. - Omar E. Pol, Nov 09 2011
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
Inverse Stirling transform of A065143. - Vladimir Reshetnikov, Nov 01 2015
FORMULA
a(n) = (2^n + (-2)^n)/2 = 2^(n-1)*(1 + (-1)^n).
O.g.f.: 1/(1-(2*x)^2).
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
EXAMPLE
a(2) = 4 from starting with vertex no. 1, with edges e1 and e2 to vertex no. 2: e1e1, e2e2, e1e2 and e2e1.
MATHEMATICA
nn = 39; Drop[Range[0, nn]! CoefficientList[Series[ Sinh[x] Cosh[x], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jun 10 2013 *)
PROG
(PARI) vector(100, n, n--; (2^(n) +(-2)^n)/2) \\ Altug Alkan, Nov 02 2015
CROSSREFS
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.
Cf. A065143 (Stirling transform).
Sequence in context: A079986 A134746 A210067 * A003195 A190759 A086262
KEYWORD
nonn,easy,walk,changed
AUTHOR
Wolfdieter Lang, Nov 08 2011
STATUS
approved