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%I #51 Nov 08 2022 11:07:57
%S 1,8,192,12288,1966080,1509949440
%N 1/2 times the number of n X n 0..3 matrices M with MM' mod 4 = I, where M' is the transpose of M and I is the n X n identity matrix.
%C It seems that a(n) = n! * 2^(binomial(n+1,2) - 1) for n = 1, 2, 3, 4, 5, while for n = 6, a(n) is twice this number. The number n! * 2^(binomial(n+1,2) - 1) appears in Proposition 6.1 in Eriksson and Linusson (2000) as an upper bound to the number of three-dimensional permutation arrays of size n (see column k = 3 of A330490). - _Petros Hadjicostas_, Dec 16 2019
%H Kimmo Eriksson and Svante Linusson, <a href="https://doi.org/10.1006/aama.1999.0693">A combinatorial theory of higher-dimensional permutation arrays</a>, Adv. Appl. Math. 25(2) (2000a), 194-211; see Proposition 6.1 (p. 210).
%H Jianing Song, <a href="/A060968/a060968.txt">Structure of the group SO(2,Z_n)</a>.
%H László Tóth, <a href="http://arxiv.org/abs/1404.4214">Counting solutions of quadratic congruences in several variables revisited</a>, arXiv:1404.4214 [math.NT], 2014.
%H László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Toth/toth12.html">Counting Solutions of Quadratic Congruences in Several Variables Revisited</a>, J. Int. Seq. 17 (2014), #14.11.6.
%e From _Petros Hadjicostas_, Dec 16 2019: (Start)
%e For n = 2, here are the 2*a(2) = 16 2 x 2 matrices M with elements in {0,1,2,3} that satisfy MM' mod 4 = I:
%e (a) With 1 = det(M) mod 4:
%e [[1,0],[0,1]]; [[0,1],[3,0]]; [[0,3],[1,0]]; [[1,2],[2,1]];
%e [[2,1],[3,2]]; [[2,3],[1,2]]; [[3,0],[0,3]]; [[3,2],[2,3]].
%e These form the abelian group SO(2, Z_n). See the comments for sequence A060968.
%e (b) With 3 = det(M) mod 4:
%e [[0,1],[1,0]]; [[0,3],[3,0]]; [[1,0],[0,3]]; [[1,2],[2,3]];
%e [[2,1],[1,2]]; [[2,3],[3,2]]; [[3,0],[0,1]]; [[3,2],[2,1]].
%e Note that, for n = 3, we have 2*a(3) = 2*192 = 384 = A264083(4). (End)
%Y Cf. A003920, A060968, A071302, A071304, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A229136, A264083, A330490.
%K nonn,more
%O 1,2
%A _R. H. Hardin_, Jun 11 2002
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