%I A003111 M3069
%S A003111 1,1,3,19,225,3441,79259,2424195,94471089,4613520889,275148653115,
%T A003111 19686730313955,1664382756757625
%N A003111 Number of complete mappings of the cyclic group Z_{2n+1}.
%C A003111 A complete mapping of a cyclic group (Z_n,+) is a permutation f(x) of 
               Z_n such that f(0)=0 and such that f(x)-x is also a permutation.
%C A003111 a(n)=TSQ(n)/n where TSQ(n) is the number of solutions of the toroidal 
               semi-n-queen problem (A006717 is the sequence TSQ(2k-1)).
%D A003111 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003111 Anthony B. Evans, Orthomorphism Graphs of Groups, vol. 1535 of Lecture 
               Notes in Mathematics, Springer-Verlag, 1991.
%D A003111 J. Hsiang, D. F. Hsu and Y. P. Shieh, On the hardness of counting problems 
               of complete mappings, Discrete Math., 277 (2004), 87-100.
%D A003111 Lehmer, D. H.; Some properties of circulants. J. Number Theory 5 (1973), 
               43-54.
%D A003111 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 A003111 D. Novakovic, (2000) Computation of the number of complete mappings for 
               permutations. Cybernetics & System Analysis, No. 2, v. 36, pp. 244-247.
%D A003111 Y. P. Shieh, Partition strategies for #P-complete problems with applications 
               to enumerative combinatorics, PhD thesis, National Taiwan University, 
               2001.
%D A003111 Y. P. Shieh, J. Hsiang and D. F. Hsu, On the enumeration of Abelian k-complete 
               mappings, Vol. 144 of Congressus Numerantium, 2000, pp. 67-88.
%H A003111 Y. P. Shieh, <a href="http://turing.csie.ntu.edu.tw/~arping/cm">Cyclic 
               complete mappings counting problems</a>
%F A003111 Every term is odd and if n=2 mod 3 then a(n) is divisible by 3. Also 
               a(n) is asymptotically less than 0.62^n n!. [McKay, McLeod, Wanless]
%e A003111 f(x)=2x in (Z_7,+) is a complete mapping of Z_7 since that f(0)=0 and 
               that f(x)-x (=x) is also a permutation of Z_7.
%Y A003111 Cf. A006717, A071607, A071608, A071706, A006204.
%Y A003111 Sequence in context: A166380 A136652 A136504 this_sequence A126444 A001929 
               A157675
%Y A003111 Adjacent sequences: A003108 A003109 A003110 this_sequence A003112 A003113 
               A003114
%K A003111 nonn,nice
%O A003111 0,3
%A A003111 N. J. A. Sloane (njas(AT)research.att.com).
%E A003111 More terms from J. Hsiang, D. F. Hsu and Y. P. Shieh (arping(AT)turing.csie.ntu.edu.tw), 
               Jun 03 2002
%E A003111 a(12) from Yuh-Pyng Shieh (arping(AT)gmail.com), Jan 10 2006