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A002562
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Number of ways of placing n nonattacking queens on n X n board (symmetric solutions count only once).
(Formerly M0180 N0068)
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21
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1, 0, 0, 1, 2, 1, 6, 12, 46, 92, 341, 1787, 9233, 45752, 285053, 1846955, 11977939, 83263591, 621012754, 4878666808, 39333324973, 336376244042, 3029242658210, 28439272956934, 275986683743434, 2789712466510289, 29363495934315694
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,5
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REFERENCES
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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).
M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 238.
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LINKS
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Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-1. Illustrates a(8)=12.
Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-2.
Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-3.
Popular Computing (Calabasas, CA), 8 Queens, Vol. 2, No. 13, Apr 1974, page PC13-4.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47. [Incomplete annotated scan of title page and pages 18-51]
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FORMULA
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a(n) = (1/8) * (Q(n) + P(n) + 2 * R(n)), where Q(n) = A000170(n) [all solutions], P(n) = A032522(n) [point symmetric solutions] and R(n) = A033148(n) [rotationally symmetric solutions].
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EXAMPLE
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a(4) = 1:
+---------+
| . . Q . |
| Q . . . |
| . . . Q |
| . Q . . |
+---------+
a(5) = 2:
+-----------+ +-----------+
| . . . Q . | | . . . Q . |
| . Q . . . | | Q . . . . |
| . . . . Q | | . . Q . . |
| . . Q . . | | . . . . Q |
| Q . . . . | | . Q . . . |
+-----------+ +-----------+
a(6) = 1:
+-------------+
| . . . . Q . |
| . . Q . . . |
| Q . . . . . |
| . . . . . Q |
| . . . Q . . |
+-------------+
a(7) = 6:
+---------------+ +---------------+ +---------------+
| Q . . . . . . | | Q . . . . . . | | . Q . . . . . |
| . . Q . . . . | | . . . Q . . . | | . . . Q . . . |
| . . . . Q . . | | . . . . . . Q | | Q . . . . . . |
| . . . . . . Q | | . . Q . . . . | | . . . . . . Q |
| . Q . . . . . | | . . . . . Q . | | . . . . Q . . |
| . . . Q . . . | | . Q . . . . . | | . . Q . . . . |
| . . . . . Q . | | . . . . Q . . | | . . . . . Q . |
+---------------+ +---------------+ +---------------+
.
+---------------+ +---------------+ +---------------+
| . Q . . . . . | | . Q . . . . . | | . Q . . . . . |
| . . . . Q . . | | . . . . Q . . | | . . . . . Q . |
| Q . . . . . . | | . . . . . . Q | | . . Q . . . . |
| . . . Q . . . | | . . . Q . . . | | . . . . . . Q |
| . . . . . . Q | | Q . . . . . . | | . . . Q . . . |
| . . Q . . . . | | . . Q . . . . | | Q . . . . . . |
| . . . . . Q . | | . . . . . Q . | | . . . . Q . . |
+---------------+ +---------------+ +---------------+
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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a(17) and a(18) found by Ulrich Schimke in Goettingen, Germany (UlrSchimke(AT)aol.com)
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STATUS
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approved
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