A033148 Number of rotationally symmetric solutions for queens on n X n board.

1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 64, 128, 0, 0, 480, 704, 0, 0, 3328, 3264, 0, 0, 32896, 43776, 0, 0, 406784, 667904, 0, 0, 5845504, 8650752, 0, 0, 77184000, 101492736, 0, 0, 1261588480, 1795233792, 0, 0, 21517426688, 35028172800, 0, 0, 406875119616, 652044443648, 0, 0, 8613581094912, 12530550128640, 0, 0, 194409626533888, 291826098503680, 0, 0

Offset: 1

Author

Miklos SZABO (mike(AT)ludens.elte.hu)

Keywords

• nonn
• hard

Comments

From Don Knuth, Jul 17 2015: (Start)

Ahrens proved that a(n)=0 unless n=4k or 4k+1. He also proved that in the latter case, a(n) is a multiple of 2^k. He found all solutions when n was less than 20.

Kraitchik carried the calculations further (for n less than 28). In his book he tabulated only the values a(n)/2^k. He had correct entries for n=21 and n=25, but his values for n=20 and n=24 were 1 too small -- of course he had calculated everything by hand! (End)

References

• W. Ahrens, Mathematische Unterhaltungen und Spiele, 2nd edition, volume 1, Teubner, 1910, pages 249-258.
• Maurice Kraitchik, Le problème des reines, Bruxelles: L'Échiquier, 1926, page 18.

Links

• Tricia M. Brown, Kaleidoscopes, Chessboards, and Symmetry, Journal of Humanistic Mathematics, Volume 6 Issue 1 ( January 2016), pages 110-126.
• P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]
• Gheorghe Coserea, Solutions for n=20.
• Gheorghe Coserea, Solutions for n=24.
• Gheorghe Coserea, MiniZinc model for generating solutions.
• YuhPyng Shieh, Cyclic Complete Mappings Counting Problems
• M. Szabo, Non-attacking Queens Problem Page

Crossrefs

Cf. A002562, A032522, A037223, A037224, A260189.

Extensions

More terms from Jieh Hsiang and YuhPyng Shieh (arping(AT)turing.csie.ntu.edu.tw), May 20 2002