A005632 Bishops on a 2n+1 X 2n+1 board (see Robinson paper for details).

0, 0, 5, 22, 258, 1628, 18052, 145976, 1837272, 18407664, 265312848, 3184567136, 52020223648, 728304073664, 13317701313600, 213083801827200, 4314950946864000, 77669134543011584, 1725980887361498368, 34519618313219995136, 835374767116711506432, 18378244896208168541184

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence mu(2k+1).]
• R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Programs

Maple

For Maple program see A005635.

Mathematica

c[n_] := Module[{k}, If[Mod[n, 2]==0, Return[0]]; k = (n-1)/2; If[Mod[k, 2] == 0, Return[k*2^(k-1)*((k/2)!)^2], Return[2^k*(((k+1)/2)!)^2]]];
d[n_] := d[n] = If[n <= 1, 1, d[n - 1] + (n - 1)*d[n - 2]];
B[n_] := B[n] = Which[n == 0 || n == -2, 1, OddQ[n], B[n-1], True, 2*B[n-2] + (n - 2)*B[n - 4]];
S[n_] := S[n] = Module[{k}, If[Mod[n, 2]==0, 0, k = (n-1)/2; B[k]*B[k+1]]];
Q[n_] := Module[{m}, If[Mod[n, 8] != 1, Return[0]]; m = (n-1)/8; ((2*m)!)^2 /(m!)^2];
a[n_] := (c[2n+1] - S[2n+1] - Q[2n+1])/4;
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jul 23 2022, after Maple program in A005635 *)

Extensions

More terms from N. J. A. Sloane, Sep 28 2006