A202654 Number of ways to place 3 nonattacking semi-queens on an n X n board.

0, 0, 3, 52, 370, 1620, 5285, 14168, 33012, 69240, 133815, 242220, 415558, 681772, 1076985, 1646960, 2448680, 3552048, 5041707, 7018980, 9603930, 12937540, 17184013, 22533192, 29203100, 37442600, 47534175, 59796828, 74589102, 92312220, 113413345, 138388960

Offset: 1

Author

Vaclav Kotesovec, Dec 22 2011

Keywords

• nonn

Comments

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
• V. Kotesovec, Non-attacking chess pieces
• Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Crossrefs

Cf. A099152, A047659, A103220, A202655, A202656, A202657.

Programs

Mathematica

Rest@ CoefficientList[Series[-x^3*(17 x^3 + 69 x^2 + 31 x + 3)/(x - 1)^7, {x, 0, 32}], x] (* Michael De Vlieger, Aug 19 2019 *)

Formula

a(n) = 1/6*(n-2)*(n-1)*n*(n^3-5*n^2+8*n-3).

G.f.: -x^3*(17*x^3 + 69*x^2 + 31*x + 3)/(x-1)^7.