A053290 -id:A053290 - cp's OEIS frontend

A047656 a(n) = 3^((n^2-n)/2).

1, 1, 3, 27, 729, 59049, 14348907, 10460353203, 22876792454961, 150094635296999121, 2954312706550833698643, 174449211009120179071170507, 30903154382632612361920641803529, 16423203268260658146231467800709255289, 26183890704263137277674192438430182020124347

Offset: 0

N. J. A. Sloane

The number of outcomes of a chess tournament with n players.
For n >= 1, a(n) is the size of the Sylow 3-subgroup of the Chevalley group A_n(3) (sequence A053290). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
The number of binary relations on an n-element set that are both reflexive and antisymmetric. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
The sequence a(n+1) = [1,3,27,729,59049,14348907,...] is the Hankel transform (see A001906 for definition) of A047891 = 1, 3, 12, 57, 300, 1586, 9912, ... . - Philippe Deléham, Aug 29 2006
a(n) is the number of binary relations on a set with n elements that are total relations, i.e., for a relation on a set X it holds for all a and b in X that a~b or b~a (or both). E.g., a(2) = 3 because there are three total relations on a set with two elements: {(a,a),(a,b),(b,a),(b,b)}, {(a,a),(a,b),(b,b)}, and {(a,a),(b,a),(b,b)}. - Geoffrey Critzer, May 23 2008
The number of semicomplete digraphs (or weak tournaments) on n labeled nodes. - Rémy-Robert Joseph, Nov 12 2012
The number of n X n binary matrices A that have a(i,j)=0 whenever a(j,i)=1 for i!=j and zeros on the diagonal. We need only consider the (n^2-n)/2 non-diagonal entry pairs . Since each pair is of the form <0,0>, <0,1>, or <1,0>, a(n) = 3^((n^2-n)/2). - Dennis P. Walsh, Apr 03 2014
a(n) is the number of symmetric (-1,0,1)-matrices of dimension (n-1) X (n-1). - Eric W. Weisstein, Jan 03 2021

			The a(2)=3 binary 2 X 2 matrices are [0 0; 0 0], [0 1; 0 0], and [0 0; 1 0]. - _Dennis P. Walsh_, Apr 03 2014

P. A. MacMahon, Chess tournaments and the like treated by the calculus of symmetric functions, Coll. Papers I, MIT Press, 344-375.

Vincenzo Librandi, Table of n, a(n) for n = 0..65
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
T. R. Hoffman and J. P. Solazzo, Complex Two-Graphs via Equiangular Tight Frames, arXiv preprint arXiv:1408.0334 [math.CO], 2014.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Eric Weisstein's World of Mathematics, (-1,0,1)-Matrix
Eric Weisstein's World of Mathematics, Symmetric Matrix
Index entries for sequences related to tournaments

Cf. A007747.

seq(3^binomial(n, 2), n=0..12); # Zerinvary Lajos, Jun 16 2007
seq(3^((n^2-n)/2), n=0..14);

Table[3^((n^2 - n)/2), {n, 0, 14}] (* Eric W. Weisstein, Jan 03 2021 *)
3^Table[Binomial[n, 2], {n, 0, 14}] (* Eric W. Weisstein, Jan 03 2021 *)
3^Binomial[Range[0, 14], 2] (* Eric W. Weisstein, Jan 03 2021 *)
Table[Count[Tuples[{-1, 0, 1}, {n, n}], ?SymmetricMatrixQ], {n, 3}] (* _Eric W. Weisstein, Jan 03 2021 *)

a(n)=3^binomial(n+1,2) \\ Charles R Greathouse IV, Apr 17 2012

a(n+1) is the determinant of an n X n matrix M_(i, j) = C(3*i,j). - Benoit Cloitre, Aug 27 2003
Sequence is given by the Hankel transform (see A001906 for definition) of A007564 = {1, 1, 4, 19, 100, 562, 3304, ...}; example: det([1, 1, 4, 19; 1, 4, 19, 100; 4, 19, 100, 562; 19, 100, 562, 3304]) = 3^6 = 729. - Philippe Deléham, Aug 20 2005
The sequence a(n+1) = [1,3,27,729,59049,14348907,...] is the Hankel transform (see A001906 for definition) of A047891 = 1, 3, 12, 57, 300, 1586, 9912, ... . - Philippe Deléham, Aug 29 2006
a(n) = 3^binomial(n,2). - Zerinvary Lajos, Jun 16 2007
G.f. A(x) satisfies: A(x) = 1 + x * A(3*x). - Ilya Gutkovskiy, Jun 04 2020
a(n) = a(n-1)*3^(n-1), a(0) = 1. - Mehdi Naima, Mar 09 2022

A053291 Nonsingular n X n matrices over GF(4).

Original entry on oeis.org

1, 3, 180, 181440, 2961100800, 775476766310400, 3251791214634074112000, 218210695042457748180566016000, 234298374547168764346587444978647040000, 4025200069765920285793155323595159699896401920000, 1106437515026051855463365435310419366987397763763798016000000

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..40

Cf. A002884, A003788, A027637, A053763, A053290, A053292, A053293.

Cf. A100221, A115490.

[1] cat [&*[(4^n - 4^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013

Table[Product[4^n - 4^k, {k,0,n-1}], {n,0,10}] (* Geoffrey Critzer, Jan 26 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 4^n - 4^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = (4^n - 1)*(4^n - 4)*...*(4^n - 4^(n-1)).
a(n) = A053763(n)*A027637(n). - Bruno Berselli, Jan 30 2013
From Amiram Eldar, Jul 06 2025: (Start)
a(n) = Product_{k=1..n} A115490(k).
a(n) ~ c * 4^(n^2), where c = A100221. (End)

More terms from Vladeta Jovovic, Mar 16 2000

A003787 Order of universal Chevalley group A_n (3).

Original entry on oeis.org

1, 24, 5616, 12130560, 237783237120, 42064805779476480, 67034222101339041669120, 961721214905722855895197286400, 124190524600592082795473760093457612800, 144339416867688029764487130056208182629053235200

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.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Robert Steinberg, Lectures on Chevalley Groups, Dept. of Mathematics, Yale University, 1967, pp. 130-131.
Index entries for sequences related to groups.

Cf. A003788, A003789, A003790, A003791, A003792, A053290, A100220.

[&*[(3^n - 3^k): k in [0..n-1]]/2: n in [1..10]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}];
f[3, #] & /@ Range[0, 9] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053290(n)/2. - Ralf Stephan, Mar 30 2004
a(n) = A(3,n) where A(q,n) = q^(n*(n+1)/2) * Product_{k=2..n+1}(q^k-1). - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 3^(n*(n+2)), where c = (3/2) * A100220 = 0.840189116891... . - Amiram Eldar, Jul 07 2025

One more term from Sean A. Irvine, Sep 18 2015

A053292 Number of nonsingular n X n matrices over GF(5).

Original entry on oeis.org

1, 4, 480, 1488000, 116064000000, 226614960000000000, 11064475422000000000000000, 13506266841692625000000000000000000, 412177498341354683437500000000000000000000000

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..35
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=5 of A316622.

Cf. A002884, A003789, A027872, A053290, A053291, A053293, A109345, A100222.

[1] cat [&*[(5^n - 5^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013

Table[Product[5^n - 5^k, {k,0,n-1}], {n,0,10}] (* Geoffrey Critzer, Jan 26 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 5^n - 5^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = (5^n - 1)*(5^n - 5)*...*(5^n - 5^(n-1)).
a(n) = A109345(n)*A027872(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 5^(n^2), where c = A100222. - Amiram Eldar, Jul 06 2025

More terms from Vladeta Jovovic, Mar 16 2000

A053293 Number of nonsingular n X n matrices over GF(7).

Original entry on oeis.org

1, 6, 2016, 33784128, 27811094169600, 1122211189922928537600, 2218959336124989671614429593600, 214992513152176999576908105619651923148800, 1020690003311610463765638355505358381593396977336320000, 237443634207909205360438080389756681126654524500073656592021585920000

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..30
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=7 of A316622.

Cf. A002884, A003790, A027875, A053290, A053291, A053292, A109493, A132035.

[1] cat [&*[(7^n - 7^k): k in [0..n-1]]: n in [1..7]]; // Bruno Berselli, Jan 28 2013

Table[Product[7^n - 7^k, {k, 0, n-1}], {n, 0, 10}] (* Vincenzo Librandi, Jan 28 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 7^n - 7^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = (7^n - 1)*(7^n - 7)*...*(7^n - 7^(n-1)).
a(n) = A109493(n)*A027875(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 7^(n^2), where c = A132035. - Amiram Eldar, Jul 06 2025

More terms from Vladeta Jovovic, Mar 16 2000

A060722 a(n) = 3^(n^2).

Original entry on oeis.org

1, 3, 81, 19683, 43046721, 847288609443, 150094635296999121, 239299329230617529590083, 3433683820292512484657849089281, 443426488243037769948249630619149892803

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001

The number of n X n {-1,0,1}-matrices (the same as the number of n X n matrices over GF(3) ).

Harry J. Smith, Table of n, a(n) for n = 0..45
Robert Corless and Steven Thornton, The Bohemian Eigenvalue Project, 2017 poster.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.

Cf. A053290, A053764, A378666.

for n from 1 to 15 do printf(`%d,`,3^(n^2)) od:

Array[3^(#^2) &, 9] (* Michael De Vlieger, Jul 12 2018 *)

{ for (n=0, 45, write("b060722.txt", n, " ", 3^(n^2)); ) } \\ Harry J. Smith, Jul 10 2009

a(n) = 3^(n^2); \\ Joerg Arndt, Feb 23 2014

a(n) = [x^n] 1/(1 - 3^n*x). - Ilya Gutkovskiy, Oct 10 2017
From Geoffrey Critzer, Dec 02 2024: (Start)
a(n) = Sum_{k=0..n} A378666(n,k)*A053764(k)*A053290(n-k).
Sum_{n>=0} a(n)x^n/B(n) = f(x)*g(x) where f(x) = Sum_{n>=0} A053290(n)x^n/B(n) and g(x) = Sum_{n>=0} A053764(n)x^n/B(n) and B(n) = A053290(n)/2^n. (End)

More terms from James Sellers, Apr 24 2001

a(0) = 1 added by N. J. A. Sloane, Nov 23 2007

A015001 q-factorial numbers for q=3.

Original entry on oeis.org

1, 1, 4, 52, 2080, 251680, 91611520, 100131391360, 328430963660800, 3232089113385932800, 95424198983606279987200, 8452007576574959037306265600, 2245867453247498115393020895232000, 1790317944898228845164815929864036352000

Offset: 0

Olivier Gérard

a(n) is the number of maximal chains in the lattice of subspaces of an n-dimensional vector space over GF(3). - Geoffrey Critzer, Sep 07 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Index entries for sequences related to factorial numbers.

Cf. A015002, A015004, A015005, A015006, A015007, A015008, A015009, A015011.
Column q=3 of A069777.
Cf. A000079, A003462, A047656, A053290, A100220.

[n le 1 select 1 else (3^n-1)*Self(n-1)/2: n in [1..15]]; // Vincenzo Librandi, Oct 22 2012

RecurrenceTable[{a[1]==1, a[n]==((3^n - 1) * a[n-1])/2}, a, {n,15}] (* Vincenzo Librandi, Oct 27 2012 *)
Table[QFactorial[n, 3], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} (q^k - 1) / (q - 1).
a(0) = 1, a(n) = (3^n - 1)*a(n-1)/2. - Vincenzo Librandi, Oct 27 2012
a(n) = (Product_{i=0..n-1} (q^n-q^i))/((q-1)^n*q^binomial(n,2)) = A053290(n)/(A000079(n)*A047656(n)). - Geoffrey Critzer, Sep 07 2022
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A003462(k).
a(n) ~ c * 3^(n*(n+1)/2)/2^n, where c = A100220. (End)

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

A316622 Array read by antidiagonals: T(n,k) is the order of the group GL(n,Z_k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 2, 48, 168, 1, 1, 4, 96, 11232, 20160, 1, 1, 2, 480, 86016, 24261120, 9999360, 1, 1, 6, 288, 1488000, 1321205760, 475566474240, 20158709760, 1, 1, 4, 2016, 1886976, 116064000000, 335522845163520, 84129611558952960, 163849992929280, 1

Offset: 0

Andrew Howroyd, Jul 08 2018

All rows are multiplicative.
Equivalently, the number of invertible n X n matrices mod k.
Also, for k prime (but not higher prime powers) the number of nonsingular n X n matrices over GF(k).
For k >= 2, n! divides T(n,k) since the subgroup of GL(n,k) consisting of all permutation matrices is isomorphic to S_n (the n-th symmetric group). Note that a permutation matrix is an orthogonal matrix, hence having determinant +-1. - Jianing Song, Oct 29 2022

			Array begins:
=================================================================
n\k| 1       2         3          4             5           6
---+-------------------------------------------------------------
0  | 1       1         1          1            1            1 ...
1  | 1       1         2          2            4            2 ...
2  | 1       6        48         96          480          288 ...
3  | 1     168     11232      86016      1488000      1886976 ...
4  | 1   20160  24261120 1321205760 116064000000 489104179200 ...
5  | 1 9999360  ...
...

R. P. Brent and B. D. McKay, Determinants and ranks of random matrices over Zm, Discrete Mathematics 66 (1987) pp. 35-49.
J. M. Lockhart and W. P. Wardlaw, Determinants of Matrices over the Integers Modulo m, Mathematics Magazine, Vol. 80, No. 3 (Jun., 2007), pp. 207-214.
The Group Properties Wiki, Order formulas for linear groups

Rows n=2..4 are A000252, A064767, A305186.
Columns k=2..7 are A002884, A053290, A065128, A053292, A065498, A053293.
Cf. A053291 (GF(4)), A052496 (GF(8)), A052497 (GF(9)).
Cf. A316623.

T:=function(n,k) if k=1 or n=0 then return 1; else return Order(GL(n, Integers mod k)); fi; end;
for n in [0..5] do Print(List([1..6], k->T(n,k)), "\n"); od;

T[, 1] = T[0, ] = 1; T[n_, k_] := T[n, k] = Module[{f = FactorInteger[k], p, e}, If[Length[f] == 1, {p, e} = f[[1]]; (p^e)^(n^2)* Product[(1 - 1/p^j), {j, 1, n}], Times @@ (T[n, Power @@ #]& /@ f)]];
Table[T[n - k + 1, k], {n, 0, 8}, {k, n + 1, 1, -1}] // Flatten (* Jean-François Alcover, Jul 25 2019 *)

T(n,k)={my(f=factor(k)); k^(n^2) * prod(i=1, #f~, my(p=f[i,1]); prod(j=1, n, (1 - p^(-j))))}

T(n,p^e) = (p^e)^(n^2) * Product_{j=1..n} (1 - 1/p^j) for prime p.

A053846 Number of n X n matrices over GF(3) of order dividing 2 (i.e., number of solutions of X^2=I in GL(n,3)).

Original entry on oeis.org

1, 2, 14, 236, 12692, 1783784, 811523288, 995733306992, 3988947598331024, 43581058503809001248, 1559669026899267564563936, 152805492791495918971070907584, 49094725258525117931062810300451648, 43237014297639482582550110281347475757696, 124920254287369111633119733942816364074145497472

Offset: 0

Vladeta Jovovic, Mar 28 2000

nonn

Or, number of n X n invertible diagonalizable matrices over GF(3).

			a(2) = 14 because we have: {{0, 1}, {1, 0}}, {{0, 2}, {2, 0}}, {{1, 0}, {0, 1}}, {{1, 0}, {0,2}}, {{1, 0}, {1, 2}}, {{1, 0}, {2, 2}}, {{1, 1}, {0, 2}}, {{1,2}, {0, 2}}, {{2, 0}, {0, 1}}, {{2, 0}, {0, 2}}, {{2, 0}, {1,1}}, {{2, 0}, {2, 1}}, {{2, 1}, {0, 1}}, {{2, 2}, {0, 1}}. - _Geoffrey Critzer_, Aug 05 2017

Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Andrew Howroyd, Table of n, a(n) for n = 0..60
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A053722, A053290, A290516.

Row sums of A378666.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, T(n-1, k-1)+3^k*T(n-1, k)))
    end:
a:= n-> add(3^(k*(n-k))*T(n, k), k=0...n):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 06 2017

nn = 14; g[ n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[
QFactorial[n, q]] /. q -> 3; G[z_] := Sum[z^k/g[k], {k, 0, nn}];Table[g[n], {n, 0, nn}] CoefficientList[Series[G[z]^2, {z, 0, nn}], z] (* Geoffrey Critzer, Aug 05 2017 *)

a(n)={my(v=[1]); for(n=1,n,v=vector(#v+1,k,if(k>1, v[k-1]) + if(k<=#v, 3^(k-1)*v[k]))); sum(k=0,n,3^(k*(n-k))*v[k+1])} \\ Andrew Howroyd, Mar 02 2018

from sympy.core.cache import cacheit
@cacheit
def T(n, k): return 0 if k<0 or k>n else 1 if n==0 else T(n - 1, k - 1) + 3**k*T(n - 1, k)
def a(n): return sum(3**(k*(n - k))*T(n, k) for k in range(n + 1))
print([a(n) for n in range(15)]) # Indranil Ghosh, Aug 06 2017, after Maple code

a(n)/A053290(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A053290(n))^2. - Geoffrey Critzer, Aug 05 2017

More terms from Geoffrey Critzer, Aug 05 2017

A052496 Number of nonsingular n X n matrices over GF(8).

Original entry on oeis.org

1, 7, 3528, 115379712, 241909719367680, 32467582052437076213760, 278893342293098904613804037898240, 153323163270070838469523866093442017326530560

Offset: 0

Vladeta Jovovic, Mar 16 2000

G. C. Greubel, Table of n, a(n) for n = 0..30

Cf. A027876, A109966.

Cf. A002884, A003791, A053290, A053291, A053292, A053293, A052497, A052498, A132036.

[1] cat [&*[(8^n-8^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 30 2013

Table[Product[(8^n - 8^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)

{a(n) = prod(j=0,n-1, 8^n - 8^j)}; \\ G. C. Greubel, May 14 2019

[product(8^n - 8^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (8^n - 1)*(8^n - 8)*...*(8^n - 8^(n-1)).
a(n) = A109966(n)*A027876(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 8^(n^2), where c = A132036. - Amiram Eldar, Jul 06 2025

cp's OEIS Frontend

A047656 a(n) = 3^((n^2-n)/2).

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A053291 Nonsingular n X n matrices over GF(4).

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Magma

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A003787 Order of universal Chevalley group A_n (3).

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Magma

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A053292 Number of nonsingular n X n matrices over GF(5).

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Magma

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A053293 Number of nonsingular n X n matrices over GF(7).

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Magma

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A060722 a(n) = 3^(n^2).

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Maple

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A015001 q-factorial numbers for q=3.

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