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

Author

Eric W. Weisstein, Jun 19 2017

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

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]

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

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