A288918 Number of 4-cycles in the n X n king graph.

0, 3, 29, 79, 153, 251, 373, 519, 689, 883, 1101, 1343, 1609, 1899, 2213, 2551, 2913, 3299, 3709, 4143, 4601, 5083, 5589, 6119, 6673, 7251, 7853, 8479, 9129, 9803, 10501, 11223, 11969, 12739, 13533, 14351, 15193, 16059, 16949, 17863

Offset

1,2

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) = 12*n^2 - 34*n + 23 for n > 1. - Andrew Howroyd, Jun 19 2017

From Colin Barker, Mar 11 2019: (Start)

G.f.: x^2*(3 + 20*x + x^2) / (1 - x)^3.

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

E.g.f.: exp(x)*(23 - 22*x + 12*x^2) - 23 - x. - Stefano Spezia, Aug 14 2022

Mathematica

Table[If[n == 1, 0, 23 - 34 n + 12 n^2], {n, 20}]
Join[{0}, LinearRecurrence[{3, -3, 1}, {1, 3, 29}, {2, 20}]]
CoefficientList[Series[(-3 x - 20 x^2 - x^3)/(-1 + x)^3, {x, 0, 20}], x]

Prog

(PARI) a(n)=if(n, 12*n^2-10*n+1, 0) \\ Charles R Greathouse IV, Jun 19 2017
(PARI) concat(0, Vec(x^2*(3 + 20*x + x^2) / (1 - x)^3 + O(x^40))) \\ Colin Barker, Mar 11 2019

Crossrefs

Cf. A016742 (3-cycles), A288919 (5-cycles), A288920 (6-cycles).

Sequence in context: A257293 A221745 A087210 * A190942 A106943 A281712

Adjacent sequences: A288915 A288916 A288917 * A288919 A288920 A288921

Keyword

nonn,easy

Author

Eric W. Weisstein, Jun 19 2017

Status

approved