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

1,4

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).

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

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 *)

Prog

(Magma) [(n-1)*(n-2)*(n-3)*(3*n+8)/6: n in [1..50]]; // Vincenzo Librandi, May 27 2013
(SageMath) [binomial(n-1, 3)*(3*n+8) for n in (1..50)] # G. C. Greubel, Apr 28 2022

Crossrefs

Cf. A036464, A051223, A051224.

Sequence in context: A366933 A211140 A200431 * A177291 A169711 A281391

Adjacent sequences: A172197 A172198 A172199 * A172201 A172202 A172203

Keyword

nonn,easy

Author

Vaclav Kotesovec, Jan 29 2010

Status

approved