A005634 Bishops on an n X n board (see Robinson paper for details). (Formerly M3621)

0, 0, 1, 4, 28, 85, 630, 3096, 23220, 123952, 1036080, 7230828, 66349440, 500721252, 5080269600, 45925520096, 508031496000, 4919774752448, 59256847036800, 656763354386032, 8532986691801600, 100525956801641104, 1405335512577427200, 18431883446961030912

Offset

2,4

Comments

The problem of the bishops is to determine the number of inequivalent arrangements of n bishops on an n X n chessboard such that no bishop threatens another and every unoccupied square is threatened by some bishop. Two arrangements are considered equivalent if they are isomorphic by way of one of the eight symmetries of the chessboard. - Jean-François Alcover, Jul 24 2022 (after Robinson's paper).

References

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). [The sequence epsilon(n) page 212.]

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

Links

Table of n, a(n) for n=2..25.

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Maple

For Maple program see A005635.

Mathematica

e[n_] := Module[{k}, If[Mod[n, 2] == 0, k = n/2; If[Mod[k, 2] == 0, Return[(k!*(k + 2)/2)^2], Return[((k - 1)!*(k + 1)^2/2)^2]], k = (n - 1)/2; If[Mod[k, 2] == 0, Return[((k!)^2/12)*(3*k^3 + 16*k^2 + 18*k + 8)], Return[((k - 1)!*(k + 1)!/12)*(3*k^3 + 13*k^2 - k - 3)]]]];
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]]];
M[n_] := Module[{k}, If[Mod[n, 2] == 0, k = n/2; If[Mod[k, 2] == 0, Return[k!*(k + 2)/2], Return[(k - 1)!*(k + 1)^2/2]], k = (n - 1)/2; Return[d[k]*d[k + 1]]]];
a[n_] := e[n]/8 - c[n]/8 + S[n]/4 - M[n]/4;
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Jul 23 2022, after Maple program in A005635 *)

Crossrefs

Sequence in context: A153784 A030117 A361173 * A183485 A183437 A294315

Adjacent sequences: A005631 A005632 A005633 * A005635 A005636 A005637

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved