A094659 Number of closed walks of length n at a vertex of the cyclic graph on 7 nodes C_7.

1, 0, 2, 0, 6, 0, 20, 2, 70, 18, 252, 110, 924, 572, 3434, 2730, 12902, 12376, 48926, 54264, 187036, 232562, 720062, 980674, 2789164, 4086550, 10861060, 16878420, 42484682, 69242082, 166823430, 282580872, 657178982, 1148548016, 2595874468

Offset: 0

Author

Herbert Kociemba, Jun 06 2004

Keywords

• nonn
• easy

Comments

In general, a(n,m) = (2^n/m)*Sum_{k=0..m-1} cos(2*Pi*k/m)^n gives the number of closed walks of length n at a vertex of the cyclic graph on m nodes C_m.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-2).

Crossrefs

Cf. A199572 (m=2), A078008 (m=3), A199573 (m=4), A054877 (m=5), A047849 (bisection of m=6), A063376 (bisection of m=8), A094233 (m=9), A095929 (bisection of m=10), A087433 (bisection of m=12).

Programs

Mathematica

f[n_] := FullSimplify[ TrigToExp[ 2^n/7 Sum[Cos[2Pi*k/7]^n, {k, 0, 6}]]]; Table[ f[n], {n, 0, 36}] (* Robert G. Wilson v, Jun 09 2004 *)
LinearRecurrence[{1,4,-3,-2},{1,0,2,0},40] (* Harvey P. Dale, Jun 12 2014 *)

Formula

a(n) = (2^n/7)*Sum_{k=0..6} cos(2*Pi*k/7)^n.

a(n) = 7(a(n-2) - 2a(n-4) + a(n-6)) + 2a(n-7).

G.f.: (1-x-2x^2+x^3)/((2x-1)(-1-x+2x^2+x^3)).

a(0)=1, a(1)=0, a(2)=2, a(3)=0, a(n)=a(n-1)+4*a(n-2)-3*a(n-3)-2*a(n-4). - Harvey P. Dale, Jun 12 2014

7*a(n) = 2^n + 2*A094648(n). - R. J. Mathar, Nov 03 2020

Extensions

More terms from Robert G. Wilson v, Jun 09 2004