A053290 -id:A053290 - cp's OEIS frontend

A052497 Number of nonsingular n X n matrices over GF(9).

1, 8, 5760, 339655680, 1624314979123200, 629282246371356907929600, 19747506525777609095698646040576000, 50195501537943419769100848121708339934527488000

Offset: 0

Vladeta Jovovic, Mar 16 2000

G. C. Greubel, 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.

Cf. A027877, A053764.

Cf. A002884, A003792, A053290, A053291, A053292, A053293, A052496, A052498, A132037.

[1] cat [&*[(9^n - 9^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013

Table[Product[(9^n - 9^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)

{a(n) = prod(j=0,n-1, 9^n - 9^j)}; \\ G. C. Greubel, May 14 2019

[product(9^n - 9^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (9^n - 1)*(9^n - 9)*...*(9^n - 9^(n-1)).
a(n) = A053764(n)*A027877(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 9^(n^2), where c = A132037. - Amiram Eldar, Jul 06 2025

A056989 Number of nonsingular n X n (-1,0,1)-matrices (over the reals).

Original entry on oeis.org

1, 2, 48, 11808, 27947520, 609653621760, 119288919620689920

Offset: 0

Eric W. Weisstein

It would be nice to have an estimate for the asymptotic rate of growth.

			a(1) = 2: [1], [ -1].
a(2) = 48: There are 8 choices for the first column, u (say) and then the 2nd column can be anything except 0, u, -u, so 6 choices, giving a total of 8*6 = 48.

Minfeng Wang, C++ program to calculate A056989
Eric Weisstein's World of Mathematics, Nonsingular Matrix
Index entries for sequences related to binary matrices

Cf. A055165, A053290, A056990, A055165, A002884, A046747, A057981, A060722.

(* A brute force solution up to n = 4 *) a[n_] := a[n] = (m = Array[x, {n, n}]; cnt = 0; iter = {#, -1, 1}& /@ Flatten[m]; Do[ If[ Det[m] != 0, cnt++], Evaluate[ Sequence @@ iter]]; cnt); Table[ Print[a[n]]; a[n], {n, 1, 4}] (* Jean-François Alcover, Oct 11 2012 *)

a(n) = A060722(n) - A057981(n). - Alois P. Heinz, Dec 02 2019

a(4) from Winston C. Yang (winston(AT)cs.wisc.edu), Aug 27 2000
Entry revised by N. J. A. Sloane, Jan 02 2007
a(5) from Giovanni Resta, Feb 20 2009
a(0)=1 prepended by Alois P. Heinz, Dec 02 2019
a(0)-a(5) confirmed and a(6) added by Minfeng Wang, May 01 2024

A052498 Number of nonsingular n X n matrices over GF(11).

Original entry on oeis.org

1, 10, 13200, 2124276000, 41393302251840000, 97602635428252959312000000, 27847155251069188894843979022720000000, 961359275427083998992553051820498439890246400000000

Offset: 0

Vladeta Jovovic, Mar 16 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..30

Cf. A027879, A110195.

Cf. A002884, A053290, A053291, A053292, A053293, A052496, A052497, A132267.

[1] cat [&*[(11^n - 11^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013

Table[Product[11^n - 11^k, {k, 0, n-1}], {n, 0, 10}] (* Vincenzo Librandi, Jan 28 2013 *)

{a(n) = prod(j=0,n-1, 11^n - 11^j)}; \\ G. C. Greubel, May 14 2019

[product(11^n - 11^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (11^n - 1)*(11^n - 11)*...*(11^n - 11^(n-1)).
a(n) = A110195(n)*A027879(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 11^(n^2), where c = A132267. - Amiram Eldar, Jul 06 2025

A290516 Number of diagonalizable n X n matrices over GF(3).

Original entry on oeis.org

1, 3, 39, 2109, 417153, 346720179, 1233891662727, 17484682043488557, 1077565432934756756289, 290674711165255613845226787, 320439909778519092353160948081831, 1554385919734090411686737202215725913181, 33245671345010828575975932818988836416481765697

Offset: 0

Geoffrey Critzer, Aug 04 2017

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..56
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. A053290, A132186.

Row sums of A296605.

nn = 12; 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]^3, {z, 0, nn}], z]

a(n)/A053290(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A053290(n))^3.

A058502 Number of cyclic subgroups of Chevalley group A_n(3) (the group of nonsingular n X n matrices over GF(3)).

Original entry on oeis.org

2, 28, 3592, 4375990, 45881030476, 4761512868259324, 4226568802536503453176, 33913358590726454284540553032, 2539147871121449320978174414096629904, 1751590210907807486213696422942566723118774192, 10257642878603441540313639499316772578643776243183698528

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 02 2001

nonn

Sean A. Irvine, Java program (github)

Cf. A053658 (unlabeled case), A062250, A062552, A053290.

a(n) = Sum_{d} |{g element of A_n(3): order(g)=d}| / phi(d), where phi is the Euler totient function. - Sean A. Irvine, Aug 07 2022

More terms from Vladeta Jovovic, Jul 05 2001

a(6)-a(7) corrected and more terms from Sean A. Irvine, Aug 07 2022

A061350 Maximal size of Aut(G) where G is a finite Abelian group of order n.

Original entry on oeis.org

1, 1, 2, 6, 4, 2, 6, 168, 48, 4, 10, 12, 12, 6, 8, 20160, 16, 48, 18, 24, 12, 10, 22, 336, 480, 12, 11232, 36, 28, 8, 30, 9999360, 20, 16, 24, 288, 36, 18, 24, 672, 40, 12, 42, 60, 192, 22, 46, 40320, 2016, 480, 32, 72, 52, 11232, 40, 1008, 36, 28, 58, 48, 60, 30, 288

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 07 2001

a(n) is multiplicative; if n = p^m is a prime power the maximal size of Aut(G) is attained by the elementary Abelian group G =(C_p)^m and then Aut(G) is GL(m,p) and a(n) = (p^m - 1)*(p^m - p)*...*(p^m - p^(m-1)). For general n the maximum will be for the direct product of the (C_p)^m over the prime powers dividing n and then the automorphism group is the direct product of the GL(m,p).

Equivalently, maximal size of Aut(G) where G is a nilpotent group of order n. - Eric M. Schmidt, Feb 27 2013

T. D. Noe, Table of n, a(n) for n=1..1024

Cf. A059773, A002884, A053290, A053292, A053293.

A061350 := proc(n) local ans, i, j; ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*(mul(ifactors(n)[2][i][1]^ifactors(n)[2][i][2] - ifactors(n)[2][i][1]^(j - 1), j = 1..ifactors(n)[2][i][2])): od: RETURN(ans) end:

a[p_?PrimeQ] := p-1; a[1] = 1; a[n_] := Times @@ (Product[#[[1]]^#[[2]] - #[[1]]^k, {k, 0, #[[2]]-1}]& /@ FactorInteger[n]); Table[a[n], {n, 1, 63}] (* Jean-François Alcover, May 21 2012, after Maple *)

More terms from Vladeta Jovovic, Jun 12 2001

A051680 Number of n X n invertible matrices A over GF(3) such that A-I is invertible.

Original entry on oeis.org

1, 27, 6291, 13589289, 266377183929, 47123189360124723, 75095231825148137471259, 1077370264330489309698453375441, 139124702920688202983704723564457669361

Offset: 1

Vladeta Jovovic, Mar 17 2000

Cf. A002820.

a[n_] := a[n] = 3^(n-1)*((3^n-1)*a[n-1] + (-1)^n*3^((n-3)*n/2+1)); a[1] = 1; Table[a[n], {n, 1, 9}] (* Jean-François Alcover, Jan 12 2012, after formula *)

a(n) = 3^binomial(n,2)*b(n), with b(0)=1, b(n)=(3^n-1)*b(n-1)+(-1)^(n). - Vladeta Jovovic, Aug 20 2006
From Geoffrey Critzer, Oct 17 2021: (Start)
Sum_{n>=0} a(n)*u^n/A053290(n) = 1/(1-u)*Product_{r>=1} 1-u/3^r.
Limit_{n->inf} a(n)/3^(n^2) = (Product_{r>=1} 1-1/3^r)^2. (End)

A060705 Singular n X n matrices over GF(3).

Original entry on oeis.org

1, 33, 8451, 18785601, 371722135203, 65965023738046161, 105230885027939446251843, 1510241390481066772867454516481, 195045439041853604357302110432234667203

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..45

Cf. A053290.

for n from 1 to 20 do printf(`%d,`,3^(n^2) - product(3^n - 3^j, j=0..n-1)) od:

a(n)={3^(n^2) - prod(j=0, n-1, 3^n - 3^j)} \\ Harry J. Smith, Jul 09 2009

For n >= 1, a(n) = 3^(n^2) - A053290(n) = 3^(n^2) - (3^n - 1)*(3^n - 3)*...*(3^n - 3^(n-1)).

More terms from James Sellers, Apr 23 2001

A065498 Number of invertible n X n matrices mod 6 (i.e., over the ring Z_6).

Original entry on oeis.org

1, 2, 288, 1886976, 489104179200, 4755360379856486400, 1695944421638473850132889600, 21967113634648374162210646578639667200, 10286692771039109536373764545035369981946101760000, 173770439600109774111384717714984362383506603790098046648320000

Offset: 0

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 25 2001

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
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=6 of A316622.

Cf. A002884, A048651, A053290, A065128, A100220.

a[n_] := 6^(n^2)*Product[(1 - 1/2^k)*(1 - 1/3^k), { k, 1, n} ]; Table[ a[n], {n, 0, 9} ]

a(n) = 6^(n^2) * Product_{k=1..n} ((1 - 1/2^k)(1 - 1/3^k)).
a(n) = A002884(n)*A053290(n). - Geoffrey Critzer, Jan 26 2018
a(n) ~ c * 6^(n^2), where c = A048651 * A100220 = 0.161757743053... . - Amiram Eldar, Jul 06 2025

More terms from Robert G. Wilson v, Nov 28 2001

A296605 Rectangle read by rows: T(n,k) is the number of n X n diagonalizable matrices over GF(3) that have exactly k distinct eigenvalues, n >= 0, 0 <= k <= 3.

Original entry on oeis.org

1, 0, 0, 0, 0, 3, 0, 0, 0, 3, 36, 0, 0, 3, 702, 1404, 0, 3, 38070, 379080, 0, 3, 5351346, 341368830, 0, 3, 2434569858, 1231457092866, 0, 3, 2987199920970, 17481694843567584, 0, 3, 11966842794993066, 1077553466091961763220

Offset: 0

Geoffrey Critzer, Dec 16 2017

			Array begins:
  1, 0,       0,         0,
  0, 3,       0,         0,
  0, 3,      36,         0,
  0, 3,     702,      1404,
  0, 3,   38070,    379080,
  0, 3, 5351346, 341368830

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. A290516 (row sums).

nn = 8; g[ n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[
    QFactorial[n, q]] /. q -> 3; G[u_, z_] := Sum[z^k/\[Gamma][k], {k, 0, nn}] - 1 + u ; Grid[Map[Reverse,Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[G[u, z]^3, {z, 0, nn}], {z, u}]]]

T(n,k)/A053290(n) is the coefficient of y^(3-k)*x^n in the expansion of (-1 + y + Sum_{n>=0} x^n/A053290(n))^3.

cp's OEIS Frontend

A052497 Number of nonsingular n X n matrices over GF(9).

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Magma

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A056989 Number of nonsingular n X n (-1,0,1)-matrices (over the reals).

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Mathematica

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A052498 Number of nonsingular n X n matrices over GF(11).

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Magma

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PARI

Sage

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A290516 Number of diagonalizable n X n matrices over GF(3).

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Mathematica

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A058502 Number of cyclic subgroups of Chevalley group A_n(3) (the group of nonsingular n X n matrices over GF(3)).

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A061350 Maximal size of Aut(G) where G is a finite Abelian group of order n.

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Maple

Mathematica

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A051680 Number of n X n invertible matrices A over GF(3) such that A-I is invertible.

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Mathematica

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A060705 Singular n X n matrices over GF(3).

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