A071306 - OEIS

%I #19 Nov 07 2022 02:29:26

%S 1,8,336,112896,276595200,4662288691200,546914437209907200,

%T 450219964711195607040000,2596509480922336727312302080000,

%U 104784757384177668346109081238896640000,29597339316082819652234687848790174733434880000

%N a(n) = (1/2) * (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 Also, number of n X n orthogonal matrices over GF(7) with determinant 1. - _Max Alekseyev_, Nov 06 2022

%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) = 7^k * Product_{i=0..k-1} (7^(2k) - 7^(2i)); a(2k) = (7^k + (-1)^(k+1)) * Product_{i=1..k-1} (7^(2k) - 7^(2i)) (see MacWilliams, 1969). - _Max Alekseyev_, Nov 06 2022

%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)

%o (PARI) { a071306(n) = my(t=n\2); prod(i=0, t-1, 7^(2*t)-7^(2*i)) * if(n%2, 7^t, 1/(7^t+(-1)^t)); } \\ _Max Alekseyev_, Nov 06 2022

%Y Cf. A060968, A071302, A071303, 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(6) onward from _Max Alekseyev_, Nov 06 2022