A288919 Number of 5-cycles in the n X n king graph.

0, 0, 52, 176, 372, 640, 980, 1392, 1876, 2432, 3060, 3760, 4532, 5376, 6292, 7280, 8340, 9472, 10676, 11952, 13300, 14720, 16212, 17776, 19412, 21120, 22900, 24752, 26676, 28672

Offset

1,3

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, King Graph

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 36*n^2 - 128*n + 112 for n > 1. - Andrew Howroyd, Jun 19 2017

From Colin Barker, Mar 11 2019: (Start)

G.f.: 4*x^3*(13 + 5*x) / (1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. (End)

E.g.f.: 4*(exp(x)*(28 - 23*x + 9*x^2) - 28 - 5*x). - Stefano Spezia, Aug 14 2022

Mathematica

Table[If[n == 1, 0, 4 (n - 2) (9 n - 14)], {n, 30}]
Join[{0, 0}, LinearRecurrence[{3, -3, 1}, {20, 0, 52}, {3, 20}]]
CoefficientList[Series[-((4 x^2 (13 + 5 x))/(-1 + x)^3), {x, 0, 20}], x]

Prog

(PARI) a(n)=if(n<3, 0, 36*n^2-128*n+112) \\ Charles R Greathouse IV, Jun 19 2017
(PARI) concat([0, 0], Vec(4*x^3*(13 + 5*x) / (1 - x)^3 + O(x^40))) \\ Colin Barker, Mar 11 2019

Crossrefs

Cf. A016742 (3-cycles), A288918 (4-cycles), A288920 (6-cycles).

Sequence in context: A292172 A166390 A161478 * A260549 A211564 A346825

Adjacent sequences: A288916 A288917 A288918 * A288920 A288921 A288922

Keyword

nonn,easy

Author

Eric W. Weisstein, Jun 19 2017

Status

approved