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.
1, 16, 1536, 786432, 2013265920
Offset: 1
Examples
From _Petros Hadjicostas_, Dec 18 2019: (Start)
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:
(a) Matrices M with 1 = det(M) mod 8. These form the abelian group SO(2, Z_8). See the comments for sequence A060968.
(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]].
(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]].
(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]].
All four classes of matrices have the same number of elements, that is, 16 each.
Note that for n = 3 we have 4*a(3) = 4*1536 = 6144 = A264083(8). (End)
Links
- Jianing Song, Structure of the group SO(2,Z_n).
- László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv:1404.4214 [math.NT], 2014.
- László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), #14.11.6.
Crossrefs
Formula
Conjecture: a(n) = 2^(n*(n-1)/2) * A071303(n) for n >= 1. - Michel Marcus, Nov 08 2022