A071303 1/2 times the number of n X n 0..3 matrices M with MM' mod 4 = I, where M' is the transpose of M and I is the n X n identity matrix.

1, 8, 192, 12288, 1966080, 1509949440, 5411658792960

Offset

1,2

Comments

It seems that a(n) = n! * 2^(binomial(n+1,2) - 1) for n = 1, 2, 3, 4, 5, while for n = 6, a(n) is twice this number. The number n! * 2^(binomial(n+1,2) - 1) appears in Proposition 6.1 in Eriksson and Linusson (2000) as an upper bound to the number of three-dimensional permutation arrays of size n (see column k = 3 of A330490). - Petros Hadjicostas, Dec 16 2019

a(7) = 7! * 2^30. - Sean A. Irvine, Jul 11 2024

Links

Table of n, a(n) for n=1..7.

Kimmo Eriksson and Svante Linusson, A combinatorial theory of higher-dimensional permutation arrays, Adv. Appl. Math. 25(2) (2000a), 194-211; see Proposition 6.1 (p. 210).

Sean A. Irvinne, Java program (github)

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.

Example

From Petros Hadjicostas, Dec 16 2019: (Start)
For n = 2, here are the 2*a(2) = 16 2 x 2 matrices M with elements in {0,1,2,3} that satisfy MM'  mod 4 = I:
(a) With 1 = det(M) mod 4:
  [[1,0],[0,1]]; [[0,1],[3,0]]; [[0,3],[1,0]]; [[1,2],[2,1]];
  [[2,1],[3,2]]; [[2,3],[1,2]]; [[3,0],[0,3]]; [[3,2],[2,3]].
These form the abelian group SO(2, Z_n). See the comments for sequence A060968.
(b) With 3 = det(M) mod 4:
  [[0,1],[1,0]]; [[0,3],[3,0]]; [[1,0],[0,3]];  [[1,2],[2,3]];
  [[2,1],[1,2]]; [[2,3],[3,2]]; [[3,0],[0,1]];  [[3,2],[2,1]].
Note that, for n = 3, we have 2*a(3) = 2*192 = 384 = A264083(4). (End)

Crossrefs

Cf. A003920, A060968, A071302, A071304, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A229136, A264083, A330490.

Sequence in context: A058873 A052734 A003435 * A128406 A265269 A003956

Adjacent sequences: A071300 A071301 A071302 * A071304 A071305 A071306

Keyword

nonn,more

Author

R. H. Hardin, Jun 11 2002

Extensions

a(7) from Sean A. Irvine, Jul 11 2024

Status

approved