A000900 Number of solutions to the rook problem on an n X n board having a certain symmetry group (see Robinson for details).

0, 0, 0, 1, 2, 10, 28, 106, 344, 1272, 4592, 17692, 69384, 283560, 1191984, 5171512, 23087168, 105883456, 498572416, 2404766224, 11878871456, 59975885856, 309439708352, 1628919330208, 8746079933568, 47840206525056

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
• R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
• E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
• E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
• R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
• R. G. Wilson, v, Comments on the Larsen paper (no date)

Programs

Maple

For Maple program see A000903.

Mathematica

a85[n_] := Sum[ (2k)!/k!/2^k Binomial[n, 2k], {k, 0, n/2}]; a898[n_] := Sum[ 2^k*StirlingS1[n, k]*BellB[k], {k, 0, n}]; a[n_] := (a85[n] - a898[Floor[n/2]])/2; a[1] = 0; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 13 2011, after formula *)

Formula

a(n)=(A000085(n)-A000898(int(n/2)))/2

For asymptotics see the Robinson paper.

Extensions

More terms from Vladeta Jovovic, May 09 2000