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A071900 1/4 times the number of n X n 0..7 matrices with MM' mod 8 = I, where M' is the transpose of M and I is the n X n identity matrix.

%I A071900 #22 Nov 08 2022 11:07:34
%S A071900 1,16,1536,786432,2013265920
%N A071900 1/4 times the number of n X n 0..7 matrices with MM' mod 8 = I, where M' is the transpose of M and I is the n X n identity matrix.
%H A071900 Jianing Song, <a href="/A060968/a060968.txt">Structure of the group SO(2,Z_n)</a>.
%H A071900 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 A071900 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.
%F A071900 Conjecture: a(n) = 2^(n*(n-1)/2) * A071303(n) for n >= 1. - _Michel Marcus_, Nov 08 2022
%e A071900 From _Petros Hadjicostas_, Dec 18 2019: (Start)
%e A071900 For n = 2, the 4*a(2) = 64 n X n matrices M with elements in 0..7 that satisfy MM' mod 8 = I can be classified into four categories:
%e A071900 (a) Matrices M with 1 = det(M) mod 8. These form the abelian group SO(2, Z_8). See the comments for sequence A060968.
%e A071900 (b) Matrices M with 3 = det(M) mod 8. These are the elements of the left coset A*SO(2, Z_8) = {AM: M in SO(2, Z_8)}, where A = [[3,0],[0,1]].
%e A071900 (c) Matrices M with 5 = det(M) mod 8. These are the elements of the left coset B*SO(2, Z_8) = {BM: M in SO(2, Z_8)}, where B = [[5,0],[0,1]].
%e A071900 (d) Matrices M with 7 = det(M) mod 8. These are the elements of the left coset C*SO(2, Z_8) = {CM: M in SO(2, Z_8)}, where C= [[7,0],[0,1]].
%e A071900 All four classes of matrices have the same number of elements, that is, 16 each.
%e A071900 Note that for n = 3 we have 4*a(3) = 4*1536 = 6144 = A264083(8). (End)
%Y A071900 Cf. A060968, A071302, A071303, A071304, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083.
%K A071900 nonn,more
%O A071900 1,2
%A A071900 _R. H. Hardin_, Jun 12 2002