cp's OEIS Frontend

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

%I A199573 #22 Nov 05 2020 04:53:26
%S A199573 1,0,2,0,8,0,32,0,128,0,512,0,2048,0,8192,0,32768,0,131072,0,524288,0,
%T A199573 2097152,0,8388608,0,33554432,0,134217728,0,536870912,0,2147483648,0,
%U A199573 8589934592,0,34359738368,0,137438953472,0
%N A199573 Number of round trips of length n from any of the four vertices of the cycle graph C_4.
%C A199573 See the array w(N,L) and the triangle a(K,N) given in A199571.
%C A199573 Essentially the same as A103424.
%C A199573 This is A081294 and A000004 interleaved. - _Omar E. Pol_, Nov 09 2011
%H A199573 R. J. Mathar, <a href="/A102518/a102518.pdf">Counting Walks on Finite Graphs</a>, Section 1.
%H A199573 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,4).
%F A199573 a(n) =  2^(n-2)*(1+(-1)^n), n>=2, a(0)=1.
%F A199573 O.g.f.: (1-2*x^2)/(1-(2*x)^2).
%F A199573 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
%e A199573 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.
%t A199573 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 *)
%Y A199573 Cf. A078008 (N=3), A054877 (N=5), A199571.
%K A199573 nonn,easy,walk
%O A199573 0,3
%A A199573 _Wolfdieter Lang_, Nov 08 2011