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A071304 1/2 times the 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. 5
1, 4, 120, 14400, 9360000, 29016000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Even though only 6 terms are known for this sequence (as of the time of this note), our conjecture below is based on the work of Jianing Song for sequence A318609 with the mod 3 case. (Like 3, the number 5 is also a prime number.) - Petros Hadjicostas, Dec 18 2019

LINKS

Table of n, a(n) for n=1..6.

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

FORMULA

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)

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,more

AUTHOR

R. H. Hardin, Jun 11 2002

STATUS

approved

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Last modified June 19 02:43 EDT 2021. Contains 345125 sequences. (Running on oeis4.)