A144298 Number of cycles of length 3 in the queen graph associated with an n X n chessboard.

0, 0, 4, 36, 124, 320, 672, 1260, 2152, 3456, 5260, 7700, 10884, 14976, 20104, 26460, 34192, 43520, 54612, 67716, 83020, 100800, 121264, 144716, 171384, 201600, 235612, 273780, 316372, 363776, 416280, 474300, 538144, 608256, 684964, 768740, 859932

Offset

0,3

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Graph Cycle.

Eric Weisstein's World of Mathematics, Queen Graph.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Formula

G.f.: 4*x^2*(1 + 6*x + 5*x^2 + x^3 - x^4) / ((1 - x)^5*(1 + x)^2).

a(n) = A030117(n) + (3*n-1)*binomial(n,3).

Mathematica

Table[n (5 - 10 n + 2 n^2 + 2 n^3 - (-1)^n)/4, {n, 20}] (* Eric W. Weisstein, Jun 19 2017 *)
LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {0, 4, 36, 124, 320, 672, 1260}, 20] (* Eric W. Weisstein, Jun 19 2017 *)
CoefficientList[Series[(4 x (-1 - 6 x - 5 x^2 - x^3 + x^4))/((-1 + x)^5 (1 + x)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 19 2017 *)

Prog

(PARI) concat(vector(2), Vec(4*x^2*(1 + 6*x + 5*x^2 + x^3 - x^4) / ((1 - x)^5*(1 + x)^2) + O(x^30))) \\ Colin Barker, May 11 2017

Crossrefs

Cf. A156001 (4-cycles), A288916 (5-cycles), A288917 (6-cycles).

Sequence in context: A038688 A362751 A076830 * A072109 A045490 A318150

Adjacent sequences: A144295 A144296 A144297 * A144299 A144300 A144301

Keyword

nonn,easy

Author

Sergey Perepechko, Dec 04 2008

Status

approved