A030117 Number of triangles a queen can make (starting anywhere) on an n X n board.

0, 0, 4, 28, 80, 180, 332, 560, 864, 1272, 1780, 2420, 3184, 4108, 5180, 6440, 7872, 9520, 11364, 13452, 15760, 18340, 21164, 24288, 27680, 31400, 35412, 39780, 44464, 49532, 54940, 60760, 66944, 73568, 80580, 88060, 95952, 104340, 113164, 122512, 132320, 142680

Offset

0,3

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Macalester College Problem of the Week, Problem 855. Dizzying Triangles

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Formula

a(n) = 13*binomial(n, 3) + 5*binomial(n, 2) if n is odd; a(n) = 13*binomial(n,3) + 5* binomial(n, 2) - n/2 if n is even. - Harris Kwong (kwong(AT)cs.fredonia.edu)

From Sergey Perepechko, Dec 03 2008: (Start)

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

a(n) = n*((n-1)*(13*n-8)/6 - floor(n/2)).

a(n) = A144298(n) - (3*n-1)*binomial(n,3). (End)

Mathematica

A030117[n_] := n*((n - 1)*(13*n - 8)/6 - Quotient[n, 2]);
Array[A030117, 50, 0] (* Paolo Xausa, May 04 2026 *)

Crossrefs

Cf. A144298 (cycles of length 3).

Sequence in context: A085024 A085001 A153784 * A361173 A005634 A183485

Adjacent sequences: A030114 A030115 A030116 * A030118 A030119 A030120

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Erich Friedman

More terms from Paolo Xausa, May 04 2026

Status

approved