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%I #56 Nov 07 2022 02:28:06
%S 1,4,24,576,51840,13063680,9170703360,19808719257600,
%T 131569513308979200,2600339861038664908800,
%U 152915585868239728626892800,27051378802435080953011843891200,14395932257291877030764312963579904000
%N a(n) = (1/2) * (number of n X n 0..2 matrices M with MM' mod 3 = I, where M' is the transpose of M and I is the n X n identity matrix).
%C Also, number of n X n orthogonal matrices over GF(3) with determinant 1. - _Max Alekseyev_, Nov 06 2022
%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.
%H Jessie MacWilliams, <a href="https://doi.org/10.2307/2317262">Orthogonal Matrices Over Finite Fields</a>, The American Mathematical Monthly 76:2 (1969), 152-164.
%F a(2k+1) = 3^k * Product_{i=0..k-1} (3^(2k) - 3^(2i)); a(2k) = (3^k + (-1)^(k+1)) * Product_{i=1..k-1} (3^(2k) - 3^(2i)) (see MacWilliams, 1969). - _Max Alekseyev_, Nov 06 2022
%F a(n+1) = a(n) * A318609(n+1) for n >= 1. - conjectured by _Petros Hadjicostas_, Dec 18 2019; proved based on the explicit formula by _Max Alekseyev_, Nov 06 2022
%e From _Petros Hadjicostas_, Dec 17 2019: (Start)
%e For n = 2, the 2*a(2) = 8 n X n matrices M with elements in {0, 1, 2} that satisfy MM' mod 3 = I are the following:
%e (a) With 1 = det(M) mod 3:
%e [[1,0],[0,1]]; [[0,1],[2,0]]; [[0,2],[1,0]]; [[2,0],[0,2]].
%e This is the abelian group SO(2, Z_3). See the comments for sequence A060968.
%e (b) With 2 = det(M) mod 3:
%e [[0,1],[1,0]]; [[0,2],[2,0]]; [[1,0],[0,2]]; [[2,0],[0,1]].
%e Note that, for n = 3, we have 2*a(3) = 2*24 = 48 = A264083(3). (End)
%o (PARI) { a071302(n) = my(t=n\2); prod(i=0,t-1,3^(2*t)-3^(2*i)) * if(n%2,3^t,1/(3^t+(-1)^t)); } \\ _Max Alekseyev_, Nov 06 2022
%Y Cf. A003053, A003920, A060968, A071303, A071304, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083, A318609.
%K nonn
%O 1,2
%A _R. H. Hardin_, Jun 11 2002
%E Terms a(8) onward from _Max Alekseyev_, Nov 06 2022
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