A028365 -id:A028365 - cp's OEIS frontend

A258745 Order of general affine group AGL(n,2) (=A028365(n)) divided by (n+1).

1, 1, 8, 336, 64512, 53329920, 184308203520, 2621599886868480, 152122702768688332800, 35820150273699719298416640, 34112245508649716682268134604800, 131089993748184007771243790830298726400, 2029650642403883210241235064170615545004032000

Offset: 0

Max Alekseyev, Jun 08 2015

nonn

Max Alekseyev, Table of n, a(n) for n = 0..100
Abdalla G. M. Ahmed, Matt Pharr, Victor Ostromoukhov, and Hui Huang, SZ Sequences: Binary-Based (0, 2^q)-Sequences, arXiv:2505.20434 [cs.GR], 2025. See pp. 7, 12.
Putnam Exam. 1999, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.

a(n) = A028365(n) / (n+1) = 2^n * A002884(n) / (n+1) = 2^n * n! * A053601(n) / (n+1).

A002884 Number of nonsingular n X n matrices over GF(2) (order of the group GL(n,2)); order of Chevalley group A_n (2); order of projective special linear group PSL_n(2).

Original entry on oeis.org

1, 1, 6, 168, 20160, 9999360, 20158709760, 163849992929280, 5348063769211699200, 699612310033197642547200, 366440137299948128422802227200, 768105432118265670534631586896281600, 6441762292785762141878919881400879415296000, 216123289355092695876117433338079655078664339456000

Offset: 0

N. J. A. Sloane

Also number of bases for GF(2^n) over GF(2).
Also (apparently) number of n X n matrices over GF(2) having permanent = 1. - Hugo Pfoertner, Nov 14 2003
The previous comment is true because over GF(2) permanents and determinants are the same. - Joerg Arndt, Mar 07 2008
The number of automorphisms of (Z_2)^n (the direct product of n copies of Z_2). - Peter Eastwood, Apr 06 2015
Note that n! divides a(n) since the subgroup of GL(n,2) consisting of all permutation matrices is isomorphic to S_n (the n-th symmetric group). - Jianing Song, Oct 29 2022
The number of boolean operations on n bits, or quantum operations on n qubits, that can be constructed using only CNOT (controlled NOT) gates. - David Radcliffe, Jul 06 2025

			PSL_2(2) is isomorphic to the symmetric group S_3 of order 6.

Roger W. Carter, Simple groups of Lie type. Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. viii+331pp. MR0407163 (53 #10946). See page 2.
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.
K. J. Horadam, Hadamard matrices and their applications. Princeton University Press, Princeton, NJ, 2007. xiv+263 pp. See p. 132.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..57 (first 30 terms from T. D. Noe)
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Zong Duo Dai, Solomon W. Golomb, and Guang Gong, Generating all linear orthomorphisms without repetition, Discrete Math. 205 (1999), 47-55.
P. F. Duvall, Jr. and P. W. Harley, III, A note on counting matrices, SIAM J. Appl. Math., 20 (1971), 374-377.
Nataša Ilievska and Danilo Gligoroski , Error-Detecting Code Using Linear Quasigroups, ICT Innovations 2014, Advances in Intelligent Systems and Computing Volume 311, 2015, pp 309-318.
Aaron Meyerowitz & N. J. A. Sloane, Correspondence 1979.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
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.
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
Index entries for sequences related to binary matrices.
Index entries for sequences related to groups.
Index to divisibility sequences.

Column k=2 of A316622 and A316623.
Cf. A000409, A000410, A002820, A005329, A006125, A028365, A046747.
Cf. A006516, A048651, A203303. Row sums of A381854.

[1] cat [(&*[2^n -2^k: k in [0..n-1]]): n in [1..15]]; // G. C. Greubel, Aug 31 2023

# First program
A002884:= n-> mul(2^n - 2^i, i=0..n-1);
seq(A002884(n), n = 0..12);
# Second program
A002884:= n-> 2^(n*(n-1)/2) * mul( 2^i - 1, i=1..n);
seq(A002884(n), n=0..12);

Table[Product[2^n-2^i,{i,0,n-1}],{n,0,13}] (* Harvey P. Dale, Aug 07 2011 *)
Table[2^(n*(n-1)/2) QPochhammer[2, 2, n] // Abs, {n, 0, 11}] (* Jean-François Alcover, Jul 15 2017 *)

a(n)=prod(i=2,n,2^i-1)<Charles R Greathouse IV, Jan 13 2012

[product(2^n -2^j for j in range(n)) for n in range(16)] # G. C. Greubel, Aug 31 2023

a(n) = Product_{i=0..n-1} (2^n-2^i).
a(n) = 2^(n*(n-1)/2) * Product_{i=1..n} (2^i - 1).
a(n) = A203303(n+1)/A203303(n). - R. J. Mathar, Jan 06 2012
a(n) = (6*a(n-1)^2*a(n-3) - 8*a(n-1)*a(n-2)^2) / (a(n-2)*a(n-3)) for n > 2. - Seiichi Manyama, Oct 20 2016
a(n) ~ A048651 * 2^(n^2). - Vaclav Kotesovec, May 19 2020
a(n) = A006125(n) * A005329(n). - John Keith, Jun 30 2021
a(n) = Product_{k=1..n} A006516(k). - Amiram Eldar, Jul 06 2025

A020522 a(n) = 4^n - 2^n.

Original entry on oeis.org

0, 2, 12, 56, 240, 992, 4032, 16256, 65280, 261632, 1047552, 4192256, 16773120, 67100672, 268419072, 1073709056, 4294901760, 17179738112, 68719214592, 274877382656, 1099510579200, 4398044413952, 17592181850112, 70368735789056, 281474959933440

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of walks of length 2*n+2 between any two diametrically opposite vertices of the cycle graph C_8. - Herbert Kociemba, Jul 02 2004
If we consider a(4*k+2), then 2^4 == 3^4 == 3 (mod 13); 2^(4*k+2) + 3^(4*k+2) == 3^k*(4+9) == 3*0 == 0 (mod 13). So a(4*k+2) can never be prime. - Jose Brox, Dec 27 2005
If k is odd, then a(n*k) is divisible by a(n), since: a(n*k) = (2^n)^k + (3^n)^k = (2^n + 3^n)*((2^n)^(k-1) - (2^n)^(k-2) (3^n) + - ... + (3^n)^(k-1)). So the only possible primes in the sequence are a(0) and a(2^n) for n>=1. I've checked that a(2^n) is composite for 3 <= n <= 15. As with Fermat primes, a probabilistic argument suggests that there are only finitely many primes in the sequence. - Dean Hickerson, Dec 27 2005
Let x,y,z be elements from some power set P(n), i.e., the power set of a set of n elements. Define a function f(x,y,z) in the following manner: f(x,y,z) = 1 if x is a subset of y and y is a subset of z and x does not equal z; f(x,y,z) = 0 if x is not a subset of y or y is not a subset of z or x equals z. Now sum f(x,y,z) for all x,y,z of P(n). This gives a(n). - Ross La Haye, Dec 26 2005
Number of monic (irreducible) polynomials of degree 1 over GF(2^n). - Max Alekseyev, Jan 13 2006
Let P(A) be the power set of an n-element set A and B be the Cartesian product of P(A) with itself. Then a(n) = the number of (x,y) of B for which x does not equal y. - Ross La Haye, Jan 02 2008
For n>1: central terms of the triangle in A173787. - Reinhard Zumkeller, Feb 28 2010
Pronic numbers of the form: (2^n - 1)*2^n, which is the n-th Mersenne number times 2^n, see A000225 and A002378. - Fred Daniel Kline, Nov 30 2013
Indices where records of A037870 occur. - Philippe Beaudoin, Sep 03 2014
Half the total edge length for a minimum linear arrangement of a hypercube of dimension n. (See Harper's paper below for proof). - Eitan Frachtenberg, Apr 07 2017
Number of pairs in GF(2)^{n+1} whose dot product is 1. - Christopher Purcell, Dec 11 2021

			n=5: a(5) = 4^5 - 2^5 = 1024 - 32 = 992 -> '1111100000'.

Vincenzo Librandi, Table of n, a(n) for n = 0..170
M. Archibald, A. Blecher, A. Knopfmacher, and M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Tom Copeland, The Kervaire-Milnor formula
John Elias, Illustration of initial terms: Twin 2^n hexagonal numbers
L. H. Harper, Optimal Assignment of Numbers to Vertices, J. SIAM 12(1), p. 131--135, March 1964; alternative link.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
The Sixtieth William Lowell Putnam Mathematical Competition, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Ratio of successive terms of A028365.
Cf. A000225, A060867, A161168, A006516, A059153, A065442.
Cf. A000079, A265736, A020988, A047849.
Cf. A000079, A000302.

a020522 = (* 2) . a006516  -- Reinhard Zumkeller, Dec 15 2015

[4^n - 2^n: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

A020522:=n->4^n-2^n; seq(A020522(n), n=0..50); # Wesley Ivan Hurt, Nov 29 2013

Table[4^n - 2^n, {n, 40}] (* or *) LinearRecurrence[{6, -8}, {0, 2}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

a(n)=4^n-2^n \\ Charles R Greathouse IV, Jan 30 2012

def A020522(n): return (1<Chai Wah Wu, Mar 10 2025

[4^n - 2^n for n in range(0,23)] # Zerinvary Lajos, Jun 05 2009

From Herbert Kociemba, Jul 02 2004: (Start)
G.f.: 2*x/((-1 + 2*x)*(-1 + 4*x)).
a(n) = 6*a(n-1) - 8*a(n-2). (End)
E.g.f.: exp(4*x) - exp(2*x). - Mohammad K. Azarian, Jan 14 2009
From Reinhard Zumkeller, Feb 07 2006, Jaroslav Krizek, Aug 02 2009: (Start)
a(n) = A099393(n)-A000225(n+1) = A083420(n)-A099393(n).
In binary representation, n>0: n 1's followed by n 0's (A138147(n)).
A000120(a(n)) = n.
A023416(a(n)) = n.
A070939(a(n)) = 2*n.
2*a(n)+1 = A030101(A099393(n)). (End)
a(n) = A085812(n) - A001700(n). - John Molokach, Sep 28 2013
a(n) = 2*A006516(n) = A000079(n)*A000225(n) = A265736(A000225(n)). - Reinhard Zumkeller, Dec 15 2015
a(n) = (4^(n/2) - 4^(n/4))*(4^(n/2) + 4^(n/4)). - Bruno Berselli, Apr 09 2018
Sum_{n>0} 1/a(n) = E - 1, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022
a(n) = A000302(n) - A000079(n). - John Reimer Morales, Aug 04 2025

A028665 Galois numbers for p=3; order of group AGL(n,3).

Original entry on oeis.org

1, 6, 432, 303264, 1965150720, 115562653240320, 61330486826476707840, 293207687471256968260730880, 12619705781992895315056778792140800, 4888884191426907931326620039839052385484800, 17046196453240220939126401085378073952125928970649600

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..45
Index entries for sequences related to groups.

Cf. A028365, A028667, A028669, A028673, A028675, A028679, A028681, A028685.

Cf. A100220.

FoldList[ #1*3^#2 (3^#2-1)&, 1, Range[ 20 ]]
a[n_] := 3^n * Product[3^n - 3^k, {k, 0, n-1}]; Array[a, 11, 0] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 3^n * prod(k = 0, n-1, 3^n - 3^k); \\ Amiram Eldar, Jul 12 2025

a(n) = 3^n * Product_{k=0..n-1} (3^n - 3^k).

a(n) ~ c * 3^(n^2+n), where c = A100220. - Amiram Eldar, Jul 12 2025

A028667 Galois numbers for p=5; order of group AGL(n,5).

Original entry on oeis.org

1, 20, 12000, 186000000, 72540000000000, 708171750000000000000, 172882428468750000000000000000, 1055177097007236328125000000000000000000, 161006835289591673217773437500000000000000000000000, 614192019859664935862872123718261718750000000000000000000000000

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..37
Index entries for sequences related to groups.

Cf. A028365, A028665, A028669, A028673, A028675, A028679, A028681, A028685.

Cf. A100222.

FoldList[ #1*5^#2 (5^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 5^n * Product[5^n - 5^k, {k, 0, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 5^n * prod(k = 0, n-1, 5^n - 5^k); \\ Amiram Eldar, Jul 12 2025

a(n) = 5^n * Product_{k=0..n-1} (5^n - 5^k).

a(n) ~ c * 5^(n^2+n), where c = A100222. - Amiram Eldar, Jul 12 2025

A028669 Galois numbers for p=7; order of group AGL(n,7).

Original entry on oeis.org

1, 42, 98784, 11587955904, 66774437101209600, 18861003469034659931443200, 261058346935768909875766027257446400, 177055579258883302762565632026325003745732198400, 5884074751780775313126615757455645503567996488345394872320000

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..34
Index entries for sequences related to groups.

Cf. A028365, A028665, A028667, A028673, A028675, A028679, A028681, A028685.

Cf. A132035.

FoldList[ #1*7^#2 (7^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 7^n * Product[7^n - 7^k, {k, 0, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 7^n * prod(k = 0, n-1, 7^n - 7^k); \\ Amiram Eldar, Jul 12 2025

a(n) = 7^n * Product_{k=0..n-1} (7^n - 7^k).

a(n) ~ c * 7^(n^2+n), where c = A132035. - Amiram Eldar, Jul 12 2025

A028673 Galois numbers for p=11; order of group AGL(n,11).

Original entry on oeis.org

1, 110, 1597200, 2827411356000, 606039338269189440000, 15719002038355567350156912000000, 49332934203739383347738694321468865920000000, 18734172592683683919731709047397914403374452828934400000000, 860830165835516295815608223447061872667128267986790628055380728832000000000

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..30
Index entries for sequences related to groups.

Cf. A028365, A028665, A028667, A028669, A028675, A028679, A028681, A028685.

Cf. A132267.

FoldList[ #1*11^#2 (11^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 11^n * Product[11^n - 11^k, {k, 0, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 11^n * prod(k = 0, n-1, 11^n - 11^k); \\ Amiram Eldar, Jul 12 2025

a(n) = 11^n * Product_{k=0..n-1} (11^n - 11^k).

a(n) ~ c * 11^(n^2+n), where c = A132267. - Amiram Eldar, Jul 12 2025

A028675 Galois numbers for p=13; order of group AGL(n,13).

Original entry on oeis.org

1, 156, 4429152, 21368939889024, 17430690424387037091840, 2402962221899462961810522363863040, 55984406793280086114756507719510983331312762880, 220431677762305366198023742325712037545142450383991425548943360

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..29
Index entries for sequences related to groups.

Cf. A028365, A028665, A028667, A028669, A028673, A028679, A028681, A028685.

FoldList[ #1*13^#2 (13^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 13^n * Product[13^n - 13^k, {k, 0, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 13^n * prod(k = 0, n-1, 13^n - 13^k); \\ Amiram Eldar, Jul 12 2025

a(n) = 13^n * Product_{k=0..n-1} (13^n - 13^k).

a(n) ~ c * 13^(n^2+n), where c = Product_{k>=1} (1 - 1/13^k) = 0.9171624725409... . - Amiram Eldar, Jul 12 2025

A028679 Galois numbers for p=17; order of group AGL(n,17).

Original entry on oeis.org

1, 272, 22639104, 546341708980224, 3811101610741578352558080, 7683152190027081335646892427952783360, 4476375132477699824408564935442752007430598683525120, 753722313834315665863920705126825485467891025286555525186004419870720

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..28
Index entries for sequences related to groups.

Cf. A028365, A028665, A028667, A028669, A028673, A028675, A028681, A028685.

FoldList[ #1*17^#2 (17^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 17^n * Product[17^n - 17^k, {k, 0, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 17^n * prod(k = 0, n-1, 17^n - 17^k); \\ Amiram Eldar, Jul 12 2025

a(n) = 17^n * Product_{k=0..n-1} (17^n - 17^k).

a(n) ~ c * 17^(n^2+n), where c = Product_{k>=1} (1 - 1/17^k) = 0.937716969709... . - Amiram Eldar, Jul 12 2025

A028681 Galois numbers for p=19; order of group AGL(n,19).

Original entry on oeis.org

1, 342, 44446320, 2090711424299040, 35507456811518119000588800, 217698482437446717711443628892666137600, 481835288795046555242155407852974930874821656470528000, 384989616762670041552999657317996225002975219911214265653201810077696000

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..27
Index entries for sequences related to groups.

Cf. A028365, A028665, A028667, A028669, A028673, A028675, A028679, A028685.

FoldList[ #1*19^#2 (19^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 19^n * Product[19^n - 19^k, {k, 0, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 12 2025 *)

a(n) = 19^n * prod(k = 0, n-1, 19^n - 19^k); \\ Amiram Eldar, Jul 12 2025

a(n) = 19^n * Product_{k=0..n-1} (19^n - 19^k).

a(n) ~ c * 19^(n^2+n), where c = Product_{k>=1} (1 - 1/19^k) = 0.944598742929... . - Amiram Eldar, Jul 12 2025

cp's OEIS Frontend

A258745 Order of general affine group AGL(n,2) (=A028365(n)) divided by (n+1).

Views

Author

Keywords

Links

Formula

A002884 Number of nonsingular n X n matrices over GF(2) (order of the group GL(n,2)); order of Chevalley group A_n (2); order of projective special linear group PSL_n(2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A020522 a(n) = 4^n - 2^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Sage

Formula

A028665 Galois numbers for p=3; order of group AGL(n,3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A028667 Galois numbers for p=5; order of group AGL(n,5).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A028669 Galois numbers for p=7; order of group AGL(n,7).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A028673 Galois numbers for p=11; order of group AGL(n,11).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula