cp's OEIS Frontend

A095929 Number of closed walks of length 2n at a vertex of the cyclic graph on 10 nodes C_10.

%I A095929 #32 Jan 05 2023 18:43:26
%S A095929 1,2,6,20,70,254,948,3614,13990,54740,215766,854702,3396916,13530350,
%T A095929 53971350,215492564,860941798,3441074654,13757249460,55010542910,
%U A095929 219993856006,879848932052,3519064567926,14075391282830,56299295324980,225191238869774
%N A095929 Number of closed walks of length 2n at a vertex of the cyclic graph on 10 nodes C_10.
%C A095929 In general 2^n/m*Sum_{r=0..m-1} cos(2Pi*k*r/m)*cos(2Pi*r/m)^n is the number of walks of length n between two nodes at distance k in the cycle graph C_m. Here we have m=10 and k=0.
%C A095929 The number of round trips of odd length is zero. See the array w(N,L) and triangle a(K,N) given in A199571 for the general case. - _Wolfdieter Lang_, Nov 08 2011
%H A095929 Colin Barker, <a href="/A095929/b095929.txt">Table of n, a(n) for n = 0..1000</a>
%H A095929 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-13,4).
%F A095929 a(n) = 4^n/10*Sum_{r=0..9} cos(Pi*r/5)^(2*n).
%F A095929 a(n) = 7*a(n-1)-13*a(n-2)+4*a(n-3).
%F A095929 G.f.: (-1+5*x-5*x^2)/((-1+4*x)(1-3*x+x^2)).
%F A095929 a(n/2) = ( 2^n +2*phi^n +2*(phi-1)^n )*(1+(-1)^n)/10, with the golden section phi = A001622. - _Wolfdieter Lang_, Nov 08 2011
%F A095929 a(n) = (2^(-n)*(8^n+2*(3-sqrt(5))^n+2*(3+sqrt(5))^n))/5. - _Colin Barker_, Apr 27 2016
%F A095929 a(n) = (4^n + 2*Lucas(2*n))/5. - _Ehren Metcalfe_, Apr 04 2019
%F A095929 a(n) = Sum_{k=-n..n} binomial(2*n, n+5*k). - _Greg Dresden_, Jan 05 2023
%e A095929 a(2)=6 from the six round trips from, say, vertex no. 1: 12121, 1(10)1(10)1, 121(10)1, 1(10)121, 12321 and 1(10)9(10)1. - _Wolfdieter Lang_, Nov 08 2011
%t A095929 f[n_]:=FullSimplify[TrigToExp[(4^n/10)Sum[Cos[Pi*k/5]^(2n), {k, 0, 9}]]];Table[f[n], {n, 0, 35}]
%t A095929 LinearRecurrence[{7,-13,4},{1,2,6},30] (* _Harvey P. Dale_, Dec 09 2018 *)
%o A095929 (PARI) Vec((1-5*x+5*x^2)/((1-4*x)*(1-3*x+x^2)) + O(x^50)) \\ _Colin Barker_, Apr 27 2016
%Y A095929 Cf. A094233, A199571.
%K A095929 nonn,easy
%O A095929 0,2
%A A095929 _Herbert Kociemba_, Jul 12 2004