A091000 Number of closed walks of length n on the Petersen graph rooted at a given vertex.

1, 0, 3, 0, 15, 12, 99, 168, 759, 1764, 6315, 16896, 54783, 156156, 484851, 1421784, 4330887, 12861588, 38846907, 116016432, 349097871, 1045196460, 3139783683, 9410962440, 28249664535, 84715439172, 254213426379, 762506061408

Offset: 0

Author

Paul Barry, Dec 12 2003

Keywords

• easy
• nonn

Comments

If p >= 7 is a prime, then p divides a(p) (provable by easy application of Fermat's Little Theorem). - Adam P. Goucher, Sep 11 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

Programs

GAP

List([0..30], n -> (3^n+(-2)^(n+2)+5)/10); # G. C. Greubel, Feb 01 2019

Magma

[(3^n+(-2)^(n+2)+5)/10: n in [0..30]]; // G. C. Greubel, Feb 01 2019

Mathematica

Table[{1,0,0}.MatrixPower[{{0,3,0},{1,0,2},{0,1,2}},n].{1,0,0},{n,1,100}] (* Adam P. Goucher, Sep 11 2013 *)
LinearRecurrence[{2,5,-6}, {1,0,3}, 30] (* G. C. Greubel, Feb 01 2019 *)

PARI

vector(30, n, n--; (3^n+(-2)^(n+2)+5)/10) \\ G. C. Greubel, Feb 01 2019

Sage

[(3^n+(-2)^(n+2)+5)/10 for n in (0..30)] # G. C. Greubel, Feb 01 2019

Formula

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

a(n) = (3^n + (-2)^(n+2) + 5)/10.

a(n) = (A000244(n) + 9*A001045(n+1)(-1)^n + 6*A001045(n)(-1)^(n+1))/10.

3^n = a(n) + 3*A091001(n) + 6*A091002(n)

E.g.f.: (exp(3*x) + 4*exp(-2*x) + 5*exp(x))/10. - G. C. Greubel, Feb 01 2019