A091000 - OEIS

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

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OFFSET

0,3

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).

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

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 *)

PROG

(PARI) vector(30, n, 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

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

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

CROSSREFS

Sequence in context: A054882 A303232 A086479 * A361804 A387833 A389409

Adjacent sequences: A090997 A090998 A090999 * A091001 A091002 A091003

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Dec 12 2003

STATUS

approved