login
A172127
Number of ways to place 4 nonattacking bishops on an n X n board.
14
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
1,3
REFERENCES
E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63
LINKS
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv:1609.00853 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (6, -12, 2, 27, -36, 0, 36, -27, -2, 12, -6, 1).
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
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 *)
PROG
(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
(Magma) m:=50; R<x>:=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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Jan 26 2010
STATUS
approved