A288962 Number of 4-cycles in the n X n rook graph.

0, 1, 9, 60, 250, 765, 1911, 4144, 8100, 14625, 24805, 39996, 61854, 92365, 133875, 189120, 261256, 353889, 471105, 617500, 798210, 1018941, 1285999, 1606320, 1987500, 2437825, 2966301, 3582684, 4297510, 5122125, 6068715, 7150336, 8380944, 9775425, 11349625, 13120380, 15105546, 17324029, 19795815, 22542000

Offset: 1

Author

Eric W. Weisstein, Jun 20 2017

Keywords

• nonn
• easy

Links

• 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 (6, -15, 20, -15, 6, -1).

Crossrefs

Cf. A288961 (3-cycles), A288963 (5-cycles), A288960 (6-cycles).

Programs

Magma

[n^2*(n-1)*(n^2-4*n+5)/4 : n in [1..50]]; // Wesley Ivan Hurt, Apr 23 2021

Mathematica

Table[n^2 (n - 1) (n^2 - 4 n + 5)/4, {n, 20}]
Table[n Binomial[n, 2] (n^2 - 4 n + 5)/2, {n, 20}]
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 9, 60, 250, 765}, 20]
CoefficientList[Series[(x (1 + 3 x + 21 x^2 + 5 x^3))/(-1 + x)^6, {x, 0, 20}], x]

Formula

a(n) = n*binomial(n,2)*(n^2-4*n+5)/2.

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6).

G.f.: (x^2*(1+3*x+21*x^2+5*x^3))/(-1+x)^6.