A003053 -id:A003053 - 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

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

A071308 a(n) = (1/2) * (number of n X n 0..9 matrices with MM' mod 10 = I).

Original entry on oeis.org

1, 8, 720, 691200, 6739200000, 668528640000000, 663347543040000000000, 6622861869711360000000000000, 660754650163765248000000000000000000, 660543208675712843120640000000000000000000000, 6601093143555139842468151296000000000000000000000000000

Offset: 1

R. H. Hardin, Jun 11 2002

nonn

Cf. A003053, A071302, A071303, A071304, A071305, A071306, A071307, A071309, A071310.

a(n) = A003053(n) * A071304(n). - Max Alekseyev, Nov 06 2022

Terms a(6) 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

A090770 a(n) = 2^(n^2 + 2n + 1)*Product_{j=1..n} (4^j - 1).

Original entry on oeis.org

2, 48, 23040, 185794560, 24257337753600, 50821645356918374400, 1704875112338069448032256000, 915241991059360703024740763172864000, 7861748876453505095791592854589753555681280000, 1080506416218846625176535970968094253434513802154475520000, 2376056471052200653607636735377527394627947719754523173734842368000000

Offset: 0

N. J. A. Sloane, Feb 10 2004

nonn

The order of the p-Clifford group for an odd prime p is a*p^(n^2+2n+1)*Product_{j=1..n} (p^(2*j)-1), where a = gcd(p+1,4). This is the sequence obtained by (illegally) setting p = 2.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Cf. A001309, A003956, A100221.
Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7).
A bisection of A003053, cf. A003923.

Table[2^(n^2+2n+1) Product[4^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, May 14 2022 *)

from math import prod
def A090770(n): return prod((1<Chai Wah Wu, Jun 20 2022

a(n) ~ c * 2^((n+1)*(2*n+1)), where c = A100221. - Amiram Eldar, Jul 06 2025

A003923 Order of universal Chevalley group B_n (2) or symplectic group Sp(2n,2).

Original entry on oeis.org

1, 6, 720, 1451520, 47377612800, 24815256521932800, 208114637736580743168000, 27930968965434591767112450048000, 59980383884075203672726385914533642240000, 2060902435720151186326095525680721766346957783040000, 1132992015386677099994486205757869431795095310094129168384000000

Offset: 0

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

T. D. Noe, Table of n, a(n) for n = 0..20
Bernhard Runge, On Siegel modular forms. Part I, J. Reine Angew. Math., 436 (1993), 57-85.
Index entries for sequences related to groups.

A bisection of A003053.

Cf. A003920, A100221.

for m from 0 to 50 do N:=2^(m^2)*mul( 4^i-1, i=1..m); lprint(N); od:

a[n_] := 2^(n^2)*Times@@(4^Range[n]-1);
Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Aug 18 2022 *)

from math import prod
def A003923(n): return (1 << n**2)*prod((1 << i)-1 for i in range(2,2*n+1,2)) # Chai Wah Wu, Jun 20 2022

a(n) = B(2,n) where B(q,n) is defined in A003920. - Sean A. Irvine, Sep 22 2015

a(n) ~ c * 2^(n*(2*n+1)), where c = A100221. - Amiram Eldar, Jul 07 2025

Edited by N. J. A. Sloane, Dec 30 2008

A088437 Number of n X n orthogonal matrices over GF(2) modulo permutations of rows.

Original entry on oeis.org

1, 1, 1, 2, 6, 32, 288, 4608, 130560, 6684672, 621674496, 106099113984, 33421220904960, 19556188689530880, 21359269286705627136, 43743783499173124374528, 168632285389312394463805440, 1227942828363775231508883701760, 16941927202935006869128068433182720, 444122456468619444070070837134825095168

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003

nonn

Also the number of distinct self-dual bases for GF(2^n) over GF(2). - Max Alekseyev, Feb 11 2008

Max Alekseyev, PARI scripts
Joerg Arndt, Matters Computational (The Fxtbook), see p. 910.
Dieter Jungnickel, Alfred J. Menezes and Scott A. Vanstone, On the Number of Self-Dual Bases of GF(q^m) Over GF(q), Proc. Amer. Math. Soc. 109 (1990), 23-29.

Cf. A053601, A135488.

/* based on http://home.gwu.edu/~maxal/gpscripts/nsdb.gp by Max Alekseyev */
sd(m,q) =
/* Number of distinct self-dual bases of GF(q^m) over GF(q) where q is a power of prime */
{
   if ( q%2 && !(m%2), return(0) );
   return ( (q%2 + 1) * prod(i=1,m-1, q^i - (i+1)%2) / m! );
}
vector(66, n, sd(n,2)) /* Joerg Arndt, Jul 03 2011 */

a(n) = A003053(n) / n!.

More terms from Max Alekseyev, Feb 11 2008

A144545 a(n) = 2^(n(n-1))(2^n + 1)*Product_{i=1..n-1} (4^i - 1).

Original entry on oeis.org

2, 3, 60, 25920, 197406720, 25015379558400, 51615733565620224000, 1718194449153210615595008000, 918817155086936330770931156779008000, 7877103854727828347931810809383874168094720000, 1081561598265935342583934931877242782978883444539392000000

Offset: 0

N. J. A. Sloane, Dec 30 2008

nonn

T. D. Noe, Table of n, a(n) for n = 0..20

Cf. A003923, A001308, A003053, A100221.

g:=m->2^(m*(m-1))*mul( 4^i-1, i=1..m-1)*(2^m+1);

a[n_] := 2^(n*(n-1))*(2^n + 1) * Product[4^i - 1, {i, 1, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 07 2025 *)

from math import prod
def A144545(n): return ((1<Chai Wah Wu, Jun 20 2022

a(n) ~ c * 2^(2*n^2-n), where c = A100221. - Amiram Eldar, Jul 07 2025

A344674 a(n) is the maximum value such that there is an n X n binary orthogonal matrix with every row having at least a(n) ones.

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 3, 7, 5, 9, 5, 11

Offset: 1

Nathan J. Russell, May 26 2021

The inverse of an orthogonal matrix is its transpose. This implies the dot product of a row with itself must be 1. This further implies the number of ones in each row must be odd. Given that orthogonal matrices form a group, it must be the case the transpose is also an orthogonal matrix. This requires every column of a binary orthogonal matrix also have an odd number of ones. As a result, there will always be an orthogonal matrix of size n X n having rows with n-1 number of ones if n is an even number, namely an all-ones matrix except for zeros down the main diagonal. An n X n orthogonal matrix cannot exist with n-1 ones in each row if n is odd, since n-1 is even.
a(n) = n-1 if n is even.
a(n) < n-1 if n is odd.

			There exist 10 X 10 binary orthogonal matrices such that every row has at least 9 ones, but no 10 X 10 binary orthogonal matrix exists with 10 ones in each row, so a(10) = 9.
There exist 9 X 9 binary orthogonal matrices such that every row has at least 5 ones, but no 9 X 9 binary orthogonal matrix exists with 6 or more ones in each row, so a(9) = 5.

Cf. A003053.

a(11)-a(12) from Martin Ehrenstein, Jun 17 2021

A344676 The number of n X n binary orthogonal matrices having an equal number of ones in each row.

Original entry on oeis.org

1, 2, 6, 48, 120, 1440, 5040, 2903040, 203575680, 41157849600, 2414207980800

Offset: 1

Nathan J. Russell, May 26 2021

The inverse of an orthogonal matrix is its transpose. This implies the dot product of a row with itself must be 1. This further implies the number of ones in each row must be odd. Given that orthogonal matrices form a group, it must be the case the transpose is also an orthogonal matrix. This requires every column of a binary orthogonal matrix also have an odd number of ones.

For 1 <= n <= 4 the counts are the same for the total number of binary orthogonal matrices (A003053).

			a(7) = 5040. There are 5040 7 X 7 binary orthogonal matrices where all rows have an equal number of ones.

Cf. A003053.

a(9)-a(10) from Martin Ehrenstein, Jun 13 2021

a(11) from Martin Ehrenstein, Jun 16 2021

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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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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A071308 a(n) = (1/2) * (number of n X n 0..9 matrices with MM' mod 10 = I).

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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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A090770 a(n) = 2^(n^2 + 2n + 1)*Product_{j=1..n} (4^j - 1).

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A003923 Order of universal Chevalley group B_n (2) or symplectic group Sp(2n,2).

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A088437 Number of n X n orthogonal matrices over GF(2) modulo permutations of rows.

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A144545 a(n) = 2^(n*(n-1))*(2^n + 1)*Product_{i=1..n-1} (4^i - 1).

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Mathematica

A144545 a(n) = 2^(n(n-1))(2^n + 1)*Product_{i=1..n-1} (4^i - 1).