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A071306 1/2 times the number of n X n 0..6 matrices M with MM' mod 7 = I, where M' is the transpose of M and I is the n X n identity matrix. 6

%I

%S 1,8,336,112896,276595200

%N 1/2 times the number of n X n 0..6 matrices M with MM' mod 7 = I, where M' is the transpose of M and I is the n X n identity matrix.

%C Even though only 5 terms are known for this sequence, my conjecture below is based on the work (comments, formulas, etc.) of _Jianing Song_ for sequence A318609. - _Petros Hadjicostas_, Dec 19 2019

%F Conjecture: Let b(n) be the number of solutions to the equation Sum_{i = 1..n} x_i^2 = 1 (mod 7) with x_i in 0..6. We conjecture that b(n) = 7*b(n-1) - 7*b(n-2) + 49*b(n-3) for n >= 4 with b(1) = 2, b(2) = 8, and b(3) = 42. We also conjecture that a(n+1) = a(n)*b(n+1) for n >= 1. - _Petros Hadjicostas_, Dec 19 2019

%e From _Petros Hadjicostas_, Dec 19 2019: (Start)

%e For n = 2, the 2*a(2) = 16 n X n matrices M with elements in 0..6 that satisfy MM' = I are the following:

%e (a) those with 1 = det(M) mod 7:

%e [[1,0],[0,1]]; [[0,1],[6,0]]; [[0,6],[1,0]]; [[2,2],[5,2]];

%e [[2,5],[2,2]]; [[5,2],[5,5]]; [[5,5],[2,5]]; [[6,0],[0,6]].

%e These are the elements of the abelian group SO(2,Z_7). See the comments for sequence A060968.

%e (b) those with 6 = det(M) mod 7:

%e [[0,1],[1,0]]; [[0,6],[6,0]]; [[1,0],[0,6]]; [[2,2],[2,5]];

%e [[2,5],[5,5]]; [[5,2],[2,2]]; [[5,5],[5,2]]; [[6,0],[0,1]].

%e Note that, for n = 3, we have 2*a(3) = 2*336 = 672 = A264083(7). (End)

%Y Cf. A060968, A071302, A071303, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083, A318609.

%K nonn,more

%O 1,2

%A _R. H. Hardin_, Jun 11 2002

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Last modified June 22 15:24 EDT 2021. Contains 345386 sequences. (Running on oeis4.)