A172200 Number of ways to place 2 nonattacking amazons (superqueens) on an n X n board.

0, 0, 0, 20, 92, 260, 580, 1120, 1960, 3192, 4920, 7260, 10340, 14300, 19292, 25480, 33040, 42160, 53040, 65892, 80940, 98420, 118580, 141680, 167992, 197800, 231400, 269100, 311220, 358092, 410060, 467480, 530720, 600160, 676192, 759220, 849660

Offset: 1

Author

Vaclav Kotesovec, Jan 29 2010

Keywords

• nonn
• easy

Comments

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

References

• Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A036464, A051223, A051224.

Programs

Magma

[(n-1)*(n-2)*(n-3)*(3*n+8)/6: n in [1..50]]; // Vincenzo Librandi, May 27 2013

Mathematica

CoefficientList[Series[4x^3(5-2x)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,0,0,20,92},40] (* or *) Table[(n-1)(n-2)(n-3)(3n+8)/6,{n,40}] (* Harvey P. Dale, May 16 2021 *)

SageMath

[binomial(n-1,3)*(3*n+8) for n in (1..50)] # G. C. Greubel, Apr 28 2022

Formula

Explicit formula (Christian Poisson, 1990): a(n) = (n - 1)(n - 2)(n - 3)(3n + 8)/6.

G.f.: 4*x^4*(5-2*x)/(1-x)^5. - Colin Barker, Jan 09 2013

E.g.f.: 8 + (1/6)*(-48 +48*x -24*x^2 +8*x^3 +3*x^4)*exp(x). - G. C. Greubel, Apr 28 2022