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A006717 Number of ways of arranging 2n+1 nonattacking semi-queens on a (2n+1) X (2n+1) toroidal board.
(Formerly M3005)
5
1, 3, 15, 133, 2025, 37851, 1030367, 36362925, 1606008513, 87656896891, 5778121715415, 452794797220965, 41609568918940625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also the number of "good" permutations on 2n+1 elements [Novakovich]. - N. J. A. Sloane, Feb 22 2011.

Also the number of transversals of a cyclic latin square of order 2n+1 and the number of orthomorphisms of the cyclic group of order 2n+1. - Ian M. Wanless (wanless(AT)maths.ox.ac.uk), Oct 07 2001

Also the number of complete mappings of a cyclic group of order 2n+1; also (2n+1) times the number of "standard" complete mappings of cyclic group of order 2n+1. - Jieh Hsiang, D.Frank Hsu and Yuh Pyng Shieh (arping(AT)turing.csie.ntu.edu.tw), May 08 2002

See A003111 for further information.

REFERENCES

N. J. Cavenagh and I. M. Wanless, On the number of transversals in Cayley tables of cyclic groups, Disc. Appl. Math. 158 (2010), 136-146.

B. D. McKay, J. C. McLeod and I. M. Wanless, The number of transversals in a Latin square, Des. Codes Cryptogr., 40, (2006) 269-284.

D. Novakovic, (2000) Computation of the number of complete mappings for permutations. Cybernetics & System Analysis, No. 2, v. 36, pp. 244-247.

Yuh Pyng Shieh, Jieh Hsiang and D. Frank Hsu, On the enumeration of Abelian k-complete mappings, vol. 144 of Congressus Numerantium, 2000, pp. 67-88

Yuh Pyng Shieh, Partition Strategies for #P-complete problem with applications to enumerative combinatorics, PhD thesis, National Taiwan University, 2001

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

D. S. Stones and I. M. Wanless, Compound orthomorphisms of the cyclic group, Finite Fields Appl. 16 (2010), 277-289.

I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 118.

LINKS

Table of n, a(n) for n=0..12.

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013

Eric Weisstein's World of Mathematics, Queens Problem.

FORMULA

Suppose n is odd and let b(n)=a((n-1)/2). Then b(n) is odd; if n>3 and n is not 1 mod 3 then b(n) is divisible by 3n; b(n)=-2n mod n^2 in n is prime; b(n) is divisible by n^2 if n is composite; b(n) is asymptotically in between 3.2^n and 0.62^n n!. [Cavenagh, Wanless], [McKay, McLeod, Wanless], [Stones, Wanless] - Ian Wanless, Jul 30 2010

CROSSREFS

Cf. A003111, A007705.

Sequence in context: A117694 A222390 A108210 * A222263 A230166 A059861

Adjacent sequences:  A006714 A006715 A006716 * A006718 A006719 A006720

KEYWORD

nonn,more,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Jieh Hsiang, D. Frank Hsu and Yuh Pyng Shieh (arping(AT)turing.csie.ntu.edu.tw), May 08 2002

a(12) added from A003111 by N. J. A. Sloane, Mar 29 2007

Definition clarified by Vaclav Kotesovec, Sep 16 2014

STATUS

approved

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Last modified October 26 05:34 EDT 2014. Contains 248568 sequences.