A288920 Number of 6-cycles in the n X n king graph.

0, 0, 82, 430, 1030, 1882, 2986, 4342, 5950, 7810, 9922, 12286, 14902, 17770, 20890, 24262, 27886, 31762, 35890, 40270, 44902, 49786, 54922, 60310, 65950, 71842, 77986, 84382, 91030, 97930, 105082, 112486, 120142, 128050, 136210, 144622, 153286, 162202

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 (number of 3-cycles).

Cf. A288918 (number of 4-cycles).

Cf. A288919 (number of 5-cycles).

Programs

Mathematica

Table[If[n < 3, 0, 2 (275 - 267 n + 63 n^2)], {n, 20}]
CoefficientList[Series[(2 x^2 (-41 - 92 x + 7 x^2))/(-1 + x)^3, {x, 0, 20}], x]
Join[{0, 0}, LinearRecurrence[{3, -3, 1}, {142, -14, 82}, {3, 20}]]

PARI

concat(vector(2), Vec(2*x^3*(41 + 92*x - 7*x^2) / (1 - x)^3 + O(x^50))) \\ Colin Barker, Jun 19 2017

PARI

a(n)=if(n>2, 126*n^2-534*n+550, 0) \\ Charles R Greathouse IV, Jun 19 2017

Formula

From Colin Barker, Jun 19 2017: (Start)

G.f.: 2*x^3*(41 + 92*x - 7*x^2) / (1 - x)^3.

a(n) = 550 - 534*n + 126*n^2 for n > 2.

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

E.g.f.: 2*exp(x)*(275 - 204*x + 63*x^2) - 550 - 142*x + 7*x^2. - Stefano Spezia, Aug 14 2022

Extensions

More terms from Colin Barker, Jun 19 2017