A071308 -id:A071308 - cp's OEIS frontend

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

R. H. Hardin, Jun 11 2002

nonn

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

			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)

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.

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

FoldList[Times, 1, LinearRecurrence[{3, -3, 9}, {4, 6, 24}, 12]] (* Amiram Eldar, Jun 22 2025 *)

{ 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

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

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

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.

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Jun 11 2002

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

			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)

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.

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

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

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).

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Jun 11 2002

nonn

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

			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)

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).

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

{ 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

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)

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

A071306 a(n) = (1/2) * (number of n X n 0..6 matrices M with MM' mod 7 = I, where M' is the transpose of M and I is the n X n identity matrix).

Original entry on oeis.org

1, 8, 336, 112896, 276595200, 4662288691200, 546914437209907200, 450219964711195607040000, 2596509480922336727312302080000, 104784757384177668346109081238896640000, 29597339316082819652234687848790174733434880000

Offset: 1

R. H. Hardin, Jun 11 2002

nonn

Also, number of n X n orthogonal matrices over GF(7) with determinant 1. - Max Alekseyev, Nov 06 2022

			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)

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

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

{ 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

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

Terms a(6) onward from Max Alekseyev, Nov 06 2022

A071309 a(n) = (1/2) * (number of n X n 0..10 matrices with MM' mod 11 = I).

Original entry on oeis.org

1, 12, 1320, 1742400, 25721308800, 4145554781913600, 7338585441586912128000, 142998501741091915820267520000, 30655092458961006120118267244605440000, 72283553302207308288060341547889057722286080000

Offset: 1

R. H. Hardin, Jun 11 2002

nonn

Also, number of n X n orthogonal matrices over GF(11) with determinant 1. - Max Alekseyev, Nov 06 2022

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

Cf. A071302, A071303, A071304, A071305, A071306, A071307, A071308, A071310.

{ a071309(n) = my(t=n\2); prod(i=0, t-1, 11^(2*t)-11^(2*i)) * if(n%2, 11^t, 1/(11^t+(-1)^t)); } \\ Max Alekseyev, Nov 06 2022

a(2k+1) = 11^k * Product_{i=0..k-1} (11^(2k) - 11^(2i)); a(2k) = (11^k + (-1)^(k+1)) * Product_{i=1..k-1} (11^(2k) - 11^(2i)) (see MacWilliams, 1969). - Max Alekseyev, Nov 06 2022

Terms a(6) onward from Max Alekseyev, Nov 06 2022

A071305 a(n) = (1/2) * (number of n X n 0..5 matrices M with MM' mod 6 = I, where M' is the transpose of M and I is the n X n identity matrix).

Original entry on oeis.org

1, 8, 144, 27648, 37324800, 300987187200, 13311459341107200, 3680352278629318656000, 6233449457837263300853760000, 63077322283364184001573740871680000, 3794639489522011031097665950031114403840000, 1374795579913014967183977466315375129593674465280000

Offset: 1

R. H. Hardin, Jun 11 2002

nonn

			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 {0,1,2,3,4,5} that satisfy MM' mod 6 = I:
[[0,1],[1,0]]; [[0,1],[5,0]]; [[0,5],[1,0]]; [[0,5],[5,0]];
[[1,0],[0,1]]; [[1,0],[0,5]]; [[2,3],[3,2]]; [[2,3],[3,4]];
[[3,2],[2,3]]; [[3,2],[4,3]]; [[3,4],[2,3]]; [[3,4],[4,3]];
[[4,3],[3,2]]; [[4,3],[3,4]]; [[5,0],[0,1]]; [[5,0],[0,5]].
(End)

Cf. A003053, A071302, A071303, A071304, A071306, A071307, A071308, A071309, A071310.

a(n) = A003053(n) * A071302(n). - Max Alekseyev, Nov 06 2022

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

A071310 a(n) = (1/4) * (number of n X n 0..11 matrices with MM' mod 12 = I).

Original entry on oeis.org

1, 32, 4608, 7077888, 101921587200, 19725496300339200, 49628717475771816345600

Offset: 1

R. H. Hardin, Jun 11 2002

Cf. A003053, A071302, A071303, A071304, A071305, A071306, A071307, A071308, A071309, A071900.

a(n) = A071302(n) * A071303(n). - Max Alekseyev, Nov 06 2022

a(6) from Max Alekseyev, Nov 06 2022

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

A071900 1/4 times the number of n X n 0..7 matrices with MM' mod 8 = I, where M' is the transpose of M and I is the n X n identity matrix.

Original entry on oeis.org

1, 16, 1536, 786432, 2013265920

Offset: 1

R. H. Hardin, Jun 12 2002

			From _Petros Hadjicostas_, Dec 18 2019: (Start)
For n = 2, the 4*a(2) = 64 n X n matrices M with elements in 0..7 that satisfy MM' mod 8 = I can be classified into four categories:
(a) Matrices M with 1 = det(M) mod 8. These form the abelian group SO(2, Z_8). See the comments for sequence A060968.
(b) Matrices M with 3 = det(M) mod 8. These are the elements of the left coset A*SO(2, Z_8) = {AM: M in SO(2, Z_8)}, where A = [[3,0],[0,1]].
(c) Matrices M with 5 = det(M) mod 8. These are the elements of the left coset B*SO(2, Z_8) = {BM: M in SO(2, Z_8)}, where B = [[5,0],[0,1]].
(d) Matrices M with 7 = det(M) mod 8. These are the elements of the left coset C*SO(2, Z_8) = {CM: M in SO(2, Z_8)}, where C= [[7,0],[0,1]].
All four classes of matrices have the same number of elements, that is, 16 each.
Note that for n = 3 we have 4*a(3) = 4*1536 = 6144 = A264083(8). (End)

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.

Cf. A060968, A071302, A071303, A071304, A071305, A071306, A071307, A071308, A071309, A071310, A071900, A087784, A208895, A264083.

Conjecture: a(n) = 2^(n*(n-1)/2) * A071303(n) for n >= 1. - Michel Marcus, Nov 08 2022

cp's OEIS Frontend

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).

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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.

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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).

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A071306 a(n) = (1/2) * (number of n X n 0..6 matrices M with MM' mod 7 = I, where M' is the transpose of M and I is the n X n identity matrix).

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A071309 a(n) = (1/2) * (number of n X n 0..10 matrices with MM' mod 11 = I).

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A071305 a(n) = (1/2) * (number of n X n 0..5 matrices M with MM' mod 6 = I, where M' is the transpose of M and I is the n X n identity matrix).

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A071310 a(n) = (1/4) * (number of n X n 0..11 matrices with MM' mod 12 = I).

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A071900 1/4 times the number of n X n 0..7 matrices with MM' mod 8 = I, where M' is the transpose of M and I is the n X n identity matrix.

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