A172123 Number of ways to place 2 nonattacking bishops on an n X n board.

0, 4, 26, 92, 240, 520, 994, 1736, 2832, 4380, 6490, 9284, 12896, 17472, 23170, 30160, 38624, 48756, 60762, 74860, 91280, 110264, 132066, 156952, 185200, 217100, 252954, 293076, 337792, 387440, 442370, 502944, 569536, 642532

Offset: 1

Author

Vaclav Kotesovec, Jan 26 2010

Keywords

• nonn
• easy

References

• E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63

Links

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

Programs

Magma

[n*(n-1)*(3*n^2-n+2)/6: n in [1..40]]; // Vincenzo Librandi, Apr 29 2013

Magma

I:=[0, 4, 26, 92, 240]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 29 2013

Mathematica

CoefficientList[Series[-2 x (x+1)(x+2)/(x-1)^5, {x, 0, 80}], x] (* Vincenzo Librandi, Apr 29 2013 *)

PARI

a(n)=n*(n-1)*(3*n^2-n+2)/6 \\ Charles R Greathouse IV, Jun 16 2015

Formula

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

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

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Vincenzo Librandi, Apr 29 2013