A071304 a(n) = (1/2) * (number of n X n 0..4 matrices M with MM' mod 5 = I, where M' is the transpose of M and I is the n X n identity matrix).

1, 4, 120, 14400, 9360000, 29016000000, 457002000000000, 35646156000000000000, 13946558535000000000000000, 27230655539587500000000000000000, 266009466302345390625000000000000000000, 12987912192212013697265625000000000000000000000

Offset

1,2

Comments

Also, number of n X n orthogonal matrices over GF(5) with determinant 1. - Max Alekseyev, Nov 06 2022

Links

Table of n, a(n) for n=1..12.

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

Jianing Song, Structure of the group SO(2,Z_n).

Formula

a(2k+1) = 5^k * Product_{i=0..k-1} (5^(2k) - 5^(2i)); a(2k) = (5^k - 1) * Product_{i=1..k-1} (5^(2k) - 5^(2i)) (see MacWilliams, 1969). - Max Alekseyev, Nov 06 2022

From Petros Hadjicostas, Dec 20 2019: (Start)

Let b(n) be the number of solutions to the equation Sum_{i = 1..n} x_i^2 = 1 (mod 5) with x_i in 0..4. We have that b(n) = 5*b(n-1) + 5*b(n-2) - 25*b(n-3) for n >= 3 with b(0) = 0, b(1) = 2, and b(2) = 4.

We have b(n) = A330607(n, k=1) for n >= 0.

We conjecture that a(n+1) = a(n)*b(n+1) for n >= 1. (End)

Example

From Petros Hadjicostas, Dec 17 2019: (Start)
For n = 2, the 2*a(2) = 8 n X n matrices M with elements in {0,1,2,3,4} that satisfy MM' mod 5 = I are the following:
(a) those with 1 = det(M) mod 5:
[[1,0],[0,1]]; [[0,4],[1,0]]; [[0,1],[4,0]]; [[4,0],[0,4]].
These form the abelian group SO(2, Z_5). See the comments for sequence A060968.
(b) those with 4 = det(M) mod 5:
[[0,1],[1,0]]; [[0,4],[4,0]]; [[1,0],[0,4]]; [[4,0],[0,1]].
Note that, for n = 3, we have 2*a(3) = 2*120 = 240 = A264083(5). (End)

Prog

(PARI) { a071304(n) = my(t=n\2); prod(i=0, t-1, 5^(2*t)-5^(2*i)) * if(n%2, 5^t, 1/(5^t+1)); } \\ Max Alekseyev, Nov 06 2022

Crossrefs

Cf. A060968, A071302, A071303, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083, A318609, A330607.

Sequence in context: A203033 A307935 A001332 * A213957 A006607 A286757

Adjacent sequences: A071301 A071302 A071303 * A071305 A071306 A071307

Keyword

nonn

Author

R. H. Hardin, Jun 11 2002

Extensions

Terms a(7) onward from Max Alekseyev, Nov 06 2022

Status

approved