A071302 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).

1, 4, 24, 576, 51840, 13063680, 9170703360, 19808719257600, 131569513308979200, 2600339861038664908800, 152915585868239728626892800, 27051378802435080953011843891200, 14395932257291877030764312963579904000

Offset

1,2

Comments

Also, number of n X n orthogonal matrices over GF(3) with determinant 1. - Max Alekseyev, Nov 06 2022

Links

Table of n, a(n) for n=1..13.

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.

Jessie MacWilliams, Orthogonal Matrices Over Finite Fields, The American Mathematical Monthly 76:2 (1969), 152-164.

Formula

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

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

Example

From Petros Hadjicostas, Dec 17 2019: (Start)
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:
(a) With 1 = det(M) mod 3:
[[1,0],[0,1]];  [[0,1],[2,0]]; [[0,2],[1,0]]; [[2,0],[0,2]].
This is the abelian group SO(2, Z_3). See the comments for sequence A060968.
(b) With 2 = det(M) mod 3:
[[0,1],[1,0]];  [[0,2],[2,0]]; [[1,0],[0,2]]; [[2,0],[0,1]].
Note that, for n = 3, we have 2*a(3) = 2*24 = 48 = A264083(3). (End)

Prog

(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

Crossrefs

Cf. A003053, A003920, A060968, A071303, A071304, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083, A318609.

Sequence in context: A111425 A013100 A013061 * A211215 A249028 A216092

Adjacent sequences: A071299 A071300 A071301 * A071303 A071304 A071305

Keyword

nonn

Author

R. H. Hardin, Jun 11 2002

Extensions

Terms a(8) onward from Max Alekseyev, Nov 06 2022

Status

approved