OFFSET
1,2
COMMENTS
Also, number of n X n orthogonal matrices over GF(7) with determinant 1. - Max Alekseyev, Nov 06 2022
LINKS
Jessie MacWilliams, Orthogonal Matrices Over Finite Fields, The American Mathematical Monthly 76:2 (1969), 152-164.
FORMULA
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
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
EXAMPLE
From Petros Hadjicostas, Dec 19 2019: (Start)
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:
(a) those with 1 = det(M) mod 7:
[[1,0],[0,1]]; [[0,1],[6,0]]; [[0,6],[1,0]]; [[2,2],[5,2]];
[[2,5],[2,2]]; [[5,2],[5,5]]; [[5,5],[2,5]]; [[6,0],[0,6]].
These are the elements of the abelian group SO(2,Z_7). See the comments for sequence A060968.
(b) those with 6 = det(M) mod 7:
[[0,1],[1,0]]; [[0,6],[6,0]]; [[1,0],[0,6]]; [[2,2],[2,5]];
[[2,5],[5,5]]; [[5,2],[2,2]]; [[5,5],[5,2]]; [[6,0],[0,1]].
Note that, for n = 3, we have 2*a(3) = 2*336 = 672 = A264083(7). (End)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Jun 11 2002
EXTENSIONS
Terms a(6) onward from Max Alekseyev, Nov 06 2022
STATUS
approved