A199573 Number of round trips of length n from any of the four vertices of the cycle graph C_4.

1, 0, 2, 0, 8, 0, 32, 0, 128, 0, 512, 0, 2048, 0, 8192, 0, 32768, 0, 131072, 0, 524288, 0, 2097152, 0, 8388608, 0, 33554432, 0, 134217728, 0, 536870912, 0, 2147483648, 0, 8589934592, 0, 34359738368, 0, 137438953472, 0

Offset: 0

Author

Wolfdieter Lang, Nov 08 2011

Keywords

• nonn
• easy
• walk

Comments

See the array w(N,L) and the triangle a(K,N) given in A199571.

Essentially the same as A103424.

This is A081294 and A000004 interleaved. - Omar E. Pol, Nov 09 2011

Examples

			a(4)=8 from the eight round trips of length 4 (starting from, say, vertex no. 1): 12121, 14141, 12141, 14121, 12321, 14341, 12341 and 14321.
		

Links

• R. J. Mathar, Counting Walks on Finite Graphs, Section 1.
• Index entries for linear recurrences with constant coefficients, signature (0,4).

Crossrefs

Cf. A078008 (N=3), A054877 (N=5), A199571.

Programs

Mathematica

CoefficientList[Series[(1 - 2 x^2)/(1 - (2 x)^2), {x, 0, 40}], x] (* or *) Riffle[Join[{1},NestList[4#&,2,20]],0] (* or *) LinearRecurrence[ {0,4},{1,0,2},80] (* Harvey P. Dale, Dec 04 2015 *)

Formula

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

O.g.f.: (1-2*x^2)/(1-(2*x)^2).

E.g.f.: 1+(1 + 2*x^2/(U(0) - 2*x^2 + 1))*x^2 where U(k)= 4*k+5 + 2*x^2/(1 + (2*k+3)*(k+2)/U(k+1)) ; (continued fraction, 3rd kind, 2-step). - Sergei N. Gladkovskii, Oct 28 2012