

A033148


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


7



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
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OFFSET

1,4


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 249258.
Maurice Kraitchik, Le problème des reines, Bruxelles: L'Échiquier, 1926, page 18.


LINKS

Table of n, a(n) for n=1..63.
Tricia M. Brown, Kaleidoscopes, Chessboards, and Symmetry, Journal of Humanistic Mathematics, Volume 6 Issue 1 ( January 2016), pages 110126.
P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?)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, Nonattacking Queens Problem Page


CROSSREFS

Cf. A002562, A032522, A037223, A037224, A260189.
Sequence in context: A240067 A300717 A191928 * A281084 A186230 A214304
Adjacent sequences: A033145 A033146 A033147 * A033149 A033150 A033151


KEYWORD

nonn,hard


AUTHOR

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


EXTENSIONS

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


STATUS

approved



