A172141 Number of ways to place 2 nonattacking nightriders on an n X n board.

0, 6, 28, 96, 240, 518, 980, 1712, 2784, 4310, 6380, 9136, 12688, 17206, 22820, 29728, 38080, 48102, 59964, 73920, 90160, 108966, 130548, 155216, 183200, 214838, 250380, 290192, 334544, 383830, 438340, 498496, 564608, 637126

Offset: 1

Author

Vaclav Kotesovec, Jan 26 2010

Keywords

• easy
• nonn

Comments

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Christopher R. H. Hanusa, T. Zaslavsky, and S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
• 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 (3,-1,-5,5,1,-3,1).

Crossrefs

Cf. A036464, A172123, A172132, A172137.

Programs

Magma

[(n/12)*(3*(-1)^n -(11 -18*n +10*n^2 -6*n^3)): n in [1..40]]; // G. C. Greubel, Apr 21 2022

Mathematica

CoefficientList[Series[2*x*(3+2*x+x^2)*(1+x+2*x^2)/((1-x)^5*(1+x)^2), {x,0,40}], x] (* Vincenzo Librandi, May 27 2013 *)

SageMath

[(n/12)*(3*(-1)^n -(11 -18*n +10*n^2 -6*n^3)) for n in (1..40)] # G. C. Greubel, Apr 21 2022

Formula

Explicit formula (Christian Poisson, 1990): a(n) = n*(3*n^3 - 5*n^2 + 9*n - 4)/6 if n is even and a(n) = n*(n - 1)*(3*n^2 - 2*n + 7)/6 if n is odd.

G.f.: 2*x^2*(3+2*x+x^2)*(1+x+2*x^2)/((1-x)^5*(1+x)^2). - Vaclav Kotesovec, Mar 25 2010

From G. C. Greubel, Apr 21 2022: (Start)

a(n) = (1/12)*n*(3*(-1)^n - (11 - 18*n + 10*n^2 - 6*n^3)).

E.g.f.: (x/12)*(-3*exp(-x) + (3 + 30*x + 26*x^2 + 6*x^3)exp(x)). (End)