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A199572 Number of round trips of length n on the cycle graph C_2 from any of the two vertices. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

See the array and triangle A199571 for the general cycle graph C_N counting.

It appears that 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

LINKS

Table of n, a(n) for n=0..39.

Index entries for linear recurrences with constant coefficients, signature (0,4)

FORMULA

a(n) = (2^(n) +(-2)^n)/2 = 2^(n-1)*(1 + (-1)^n)), n>=0.

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, 3rd kind, 3-step). - 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

Adjacent sequences:  A199569 A199570 A199571 * A199573 A199574 A199575

KEYWORD

nonn,easy,walk

AUTHOR

Wolfdieter Lang, Nov 08 2011

STATUS

approved

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Last modified August 21 19:52 EDT 2017. Contains 290906 sequences.