A172127 Number of ways to place 4 nonattacking bishops on an n X n board.

0, 0, 8, 260, 2728, 16428, 70792, 242856, 706048, 1809464, 4199064, 8992684, 18024072, 34170724, 61784632, 107243472, 179645376, 291667440, 460615272, 709686228, 1069477928, 1579767068, 2291594536, 3269684088, 4595235136

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:1609.00853 [math.CO], 2016.
• 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 (6, -12, 2, 27, -36, 0, 36, -27, -2, 12, -6, 1).

Crossrefs

Cf. A061994, A172123, A172124.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(4*x^3*(6*x^8 +57*x^7 +316*x^6 +763*x^5 +1056*x^4 +791*x^3 +316*x^2 +53*x +2)/((1-x)^9*(x+1)^3))); // G. C. Greubel, Nov 04 2018

Mathematica

CoefficientList[Series[-4 x^2 (6 x^8 + 57 x^7 + 316 x^6 + 763 x^5 + 1056 x^4 + 791 x^3 + 316 x^2 + 53 x + 2) / ((x-1)^9 (x+1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, May 02 2013 *)
LinearRecurrence[{6,-12,2,27,-36,0,36,-27,-2,12,-6,1},{0,0,8,260,2728,16428,70792,242856,706048,1809464,4199064,8992684},30] (* Harvey P. Dale, Dec 09 2017 *)

PARI

x='x+O('x^50); concat([0,0], Vec(4*x^3*(6*x^8 +57*x^7 +316*x^6 +763*x^5 +1056*x^4 +791*x^3 +316*x^2 +53*x +2)/((1-x)^9*(x+1)^3))) \\ G. C. Greubel, Nov 04 2018

Formula

Explicit formula (Karl Fabel, 1966): a(n) = n(n - 2)(15n^6 - 90n^5 + 260n^4 - 524n^3 + 727n^2 - 646n + 348)/360 if n is even and a(n) = (n - 1)(n - 2)(15n^6 - 75n^5 + 185n^4 - 339n^3 + 388n^2 - 258n + 180)/360 if n is odd.

G.f.: 4*x^3*(6*x^8 +57*x^7 +316*x^6 +763*x^5 +1056*x^4 +791*x^3 +316*x^2 +53*x +2)/((1-x)^9*(x+1)^3). - Vaclav Kotesovec, Mar 25 2010