A003920 -id:A003920 - cp's OEIS frontend

A003927 Order of simple Chevalley group B_n (3).

25920, 4585351680, 65784756654489600, 76457792934119864313446400, 7197966128645938515382156481789952000, 54888780931741129517511777421088069718405808128000

Offset: 2

N. J. A. Sloane

nonn

Index entries for sequences related to groups.

Cf. A003920, A003938, A003939, A003940, A003941, A003942, A132037.

b[q_, n_] := q^(n^2) * Product[q^(2*k) - 1, {k, 1, n}] / GCD[2, q-1]; Table[b[3, n], {n, 2, 7}] (* Amiram Eldar, Jun 23 2025 *)

a(n) = b(3,n) where b(q,n) = q^(n^2) * Product_{k=1..n}(q^(2*k)-1) / gcd(2, q-1). - Sean A. Irvine, Sep 22 2015
a(n) = A003920(n) / 2. - Amiram Eldar, Jun 23 2025
a(n) ~ c * 3^(2*n^2+n), where c = A132037. - Amiram Eldar, Jul 09 2025

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).

Original entry on oeis.org

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

A003931 Order of universal Chevalley group B_2(q), q = prime power.

Original entry on oeis.org

720, 51840, 979200, 9360000, 276595200, 1056706560, 3443212800, 25721308800, 137037962880, 1095199948800, 2008994088960, 6114035779200, 41348052472320, 95214600000000, 205608315669120, 420206392771200, 818774509363200, 1124799322521600, 4805069329111680, 13414669637644800

Offset: 1

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.

Robin Visser, Table of n, a(n) for n = 1..10000
Index entries for sequences related to groups.

Cf. A000961, A003920, A003932, A003933, A003934, A003935, A003936, A003937.

[#SymplecticGroup(4,q) : q in [2..50] | IsPrimePower(q)];  // Robin Visser, Aug 06 2023

B[q_, n_] := q^(n^2) * Product[q^(2*k) - 1, {k, 1, n}]; Table[B[q, 2], {q, Select[Range[50], PrimePowerQ]}] (* Amiram Eldar, Jun 24 2025 *)

a(n) = B(A000961(n + 1), 2) where B(q,n) is defined in A003920. - Sean A. Irvine, Sep 22 2015

More terms from Robin Visser, Aug 06 2023

A003932 Order of universal Chevalley group B_3(q), q = prime power.

Original entry on oeis.org

1451520, 9170703360, 4106059776000, 457002000000000, 546914437209907200, 9077005607176765440, 108051462804999168000, 7338585441586912128000, 245593958671812227742720, 19266960106724096212992000, 68852034247000426560552960, 712225595693024258315904000

Offset: 1

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.

Robin Visser, Table of n, a(n) for n = 1..10000
Index entries for sequences related to groups.

Cf. A000961, A003920, A003931, A003933, A003934, A003935, A003936, A003937.

[#SymplecticGroup(6, q) : q in [2..50] | IsPrimePower(q)];  // Robin Visser, Aug 07 2023

B[q_, n_] := q^(n^2) * Product[q^(2*k) - 1, {k, 1, n}]; Table[B[q, 3], {q, Select[Range[20], PrimePowerQ]}] (* Amiram Eldar, Jun 24 2025 *)

a(n) = B(A000961(n + 1), 3) where B(q,n) is defined in A003920. - Sean A. Irvine, Sep 22 2015

More terms from Robin Visser, Aug 07 2023

A003937 Order of universal Chevalley group B_8(q), q = prime power.

Original entry on oeis.org

59980383884075203672726385914533642240000, 67806677896800158816511248022114282163091244291914415200010240000, 7084630453281025440882493116981310890142026281589018852388680249504694272000000000

Offset: 1

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.

Sean A. Irvine, Table of n, a(n) for n = 1..10
Index entries for sequences related to groups.

Cf. A000961, A003920, A003931, A003932, A003933, A003934, A003935, A003936.

B[q_, n_] := q^(n^2) * Product[q^(2*k) - 1, {k, 1, n}]; Table[B[q, 8], {q, Select[Range[10], PrimePowerQ]}] (* Amiram Eldar, Jun 24 2025 *)

a(n) = B(A000961(n + 1), 8) where B(q,n) is defined in A003920. - Sean A. Irvine, Sep 22 2015

a(3) from Sean A. Irvine, Sep 22 2015

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

A003933 Order of universal Chevalley group B_4(q), q = prime power.

Original entry on oeis.org

47377612800, 131569513308979200, 4408780839651901440000, 13946558535000000000000000, 2596509480922336727312302080000, 319368723699461283992462111539200, 22246837484597339860644476682240000

Offset: 1

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.

Index entries for sequences related to groups.

Cf. A000961, A003920, A003931, A003932, A003934, A003935, A003936, A003937.

B[q_, n_] := q^(n^2) * Product[q^(2*k) - 1, {k, 1, n}]; Table[B[q, 4], {q, Select[Range[10], PrimePowerQ]}] (* Amiram Eldar, Jun 24 2025 *)

a(n) = B(A000961(n + 1), 4) where B(q,n) is defined in A003920. - Sean A. Irvine, Sep 22 2015

a(7) from Sean A. Irvine, Sep 22 2015

A003934 Order of universal Chevalley group B_5(q), q = prime power.

Original entry on oeis.org

24815256521932800, 152915585868239728626892800, 1211875293642881119668928512000000, 266009466302345390625000000000000000000, 29597339316082819652234687848790174733434880000, 46025883638628966977843321053405598530493271244800

Offset: 1

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.

Index entries for sequences related to groups.

Cf. A000961, A003920, A003931, A003932, A003933, A003935, A003936, A003937.

B[q_, n_] := q^(n^2) * Product[q^(2*k) - 1, {k, 1, n}]; Table[B[q, 5], {q, Select[Range[10], PrimePowerQ]}] (* Amiram Eldar, Jun 24 2025 *)

a(n) = B(A000961(n + 1), 5) where B(q,n) is defined in A003920. - Sean A. Irvine, Sep 22 2015

a(5)-a(6) from Sean A. Irvine, Sep 22 2015

A003935 Order of universal Chevalley group B_6(q), q = prime power.

Original entry on oeis.org

208114637736580743168000, 14395932257291877030764312963579904000, 85278137430613949474674174708223909560320000000, 3171079936179764469273010253906250000000000000000000000

Offset: 1

N. J. A. Sloane

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.
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.

Index entries for sequences related to groups.

Cf. A000961, A003920, A003931, A003932, A003933, A003934, A003936, A003937.

B[q_, n_] := q^(n^2) * Product[q^(2*k) - 1, {k, 1, n}]; Table[B[q, 6], {q, Select[Range[10], PrimePowerQ]}] (* Amiram Eldar, Jun 24 2025 *)

a(n) = B(A000961(n + 1), 6) where B(q,n) is defined in A003920. - Sean A. Irvine, Sep 22 2015

cp's OEIS Frontend

A003927 Order of simple Chevalley group B_n (3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A003931 Order of universal Chevalley group B_2(q), q = prime power.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A003932 Order of universal Chevalley group B_3(q), q = prime power.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A003937 Order of universal Chevalley group B_8(q), q = prime power.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

Extensions