A173214 Number of ways to place 4 nonattacking amazons (superqueens) on an n X n board.

0, 0, 0, 0, 2, 112, 1754, 13074, 63400, 234014, 712248, 1882132, 4457246, 9679760, 19584514, 37367934, 67849336, 118085614, 198107620, 321870956, 508359070, 782972820, 1179105738, 1740089734, 2521359260, 3593085246, 5043058972

Offset: 1

Author

Vaclav Kotesovec, Feb 12 2010

Keywords

• nonn
• nice
• easy

Comments

A amazon (superqueen) moves like a queen and a knight.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
• Wikipedia, Amazon (chess)

Crossrefs

Cf. A172200, A172201, A061994.

Programs

Mathematica

CoefficientList[Series[2 x^4 (28 x^17 - 18 x^16 - 162 x^15 - 139 x^14 + 261 x^13 + 1268 x^12 + 2387 x^11 + 1220 x^10 - 5937 x^9 - 18637 x^8 - 30086 x^7 - 31557 x^6 - 23251 x^5 - 11716 x^4 - 3859 x^3 - 708 x^2 - 53 x - 1) / ((x + 1)^4 (x - 1)^9 (x^2 + x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

Formula

a(n) = n^8/24-5n^7/6+47n^6/9+43n^5/10-5053n^4/24+112585n^3/108-15433n^2/8+55669n/270+119917/54 + (n^3/4-21n^2/8+7n-3/2)*(-1)^n + 32/27*(n-1)*cos(2*Pi*n/3) + 40*sqrt(3)*sin(2*Pi*n/3)/81, n>=6.

Recurrence: a(n) = 3a(n-1)+a(n-2)-9a(n-3)+12a(n-5)+7a(n-6)-15a(n-7)-16a(n-8)+16a(n-9)+15a(n-10)-7a(n-11)-12a(n-12)+9a(n-14)-a(n-15)-3a(n-16)+a(n-17), n>=23. - Vaclav Kotesovec, Feb 18 2010

G.f.: 2x^5*(28x^17-18x^16-162x^15-139x^14+261x^13+1268x^12+2387x^11+1220x^10-5937x^9-18637x^8-30086x^7-31557x^6-23251x^5-11716x^4-3859x^3-708x^2-53x-1)/((x+1)^4*(x-1)^9*(x^2+x+1)^2). - Vaclav Kotesovec, Mar 24 2010