A288961 Number of 3-cycles in the n X n rook graph.

0, 0, 6, 32, 100, 240, 490, 896, 1512, 2400, 3630, 5280, 7436, 10192, 13650, 17920, 23120, 29376, 36822, 45600, 55860, 67760, 81466, 97152, 115000, 135200, 157950, 183456, 211932, 243600, 278690, 317440, 360096, 406912, 458150, 514080, 574980, 641136, 712842, 790400

Offset: 1

Author

Eric W. Weisstein, Jun 20 2017

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, Rook Graph
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A288962 (4-cycles), A288963 (5-cycles), A288960 (6-cycles).

Main diagonal of A360855.

Programs

Mathematica

Table[n^2 (n - 1) (n - 2)/3, {n, 20}]
Table[2 n Binomial[n, 3], {n, 20}]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 6, 32, 100}, 20]
CoefficientList[Series[-((2 x^2 (3 + x))/(-1 + x)^5), {x, 0, 20}], x]

PARI

a(n) = {2*n*binomial(n,3)} \\ Andrew Howroyd, Apr 26 2020

Formula

a(n) = 2*n*binomial(n,3).

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).

G.f.: (-2*x^3*(3+x))/(-1+x)^5.

Extensions

Terms a(31) and beyond from Andrew Howroyd, Apr 26 2020