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A002884
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Number of nonsingular n X n matrices over GF(2); order of Chevalley group A_n (2); order of projective special linear group PSL_n(2).
(Formerly M4302 N1798)
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65
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1, 1, 6, 168, 20160, 9999360, 20158709760, 163849992929280, 5348063769211699200, 699612310033197642547200, 366440137299948128422802227200, 768105432118265670534631586896281600
(list;
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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Carter, Roger W. 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, 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.
Horadam, K. J., 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).
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LINKS
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T. D. Noe and 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, 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.
N. Ilievska, D. Gligoroski, Error-Detecting Code Using Linear Quasigroups, ICT Innovations 2014, Advances in Intelligent Systems and Computing Volume 311, 2015, pp 309-318.
A. 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.
J. Overbey, W. Traves and J. Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Volume 29, 2005 - Issue 1.
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
Index to divisibility sequences
Index entries for sequences related to groups
Index entries for sequences related to binary matrices
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FORMULA
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Product(2^n-2^i, i=0..n-1); or 2^(n*(n-1)/2) * product( 2^i - 1, i=1..n).
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
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EXAMPLE
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PSL_2(2) is isomorphic to the symmetric group S_3 of order 6.
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MAPLE
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product(2^n-2^i, i=0..n-1); or 2^(n*(n-1)/2) * product( 2^i - 1, i=1..n);
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=prod(i=2, n, 2^i-1)<<binomial(n, 2) \\ Charles R Greathouse IV, Jan 13 2012
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CROSSREFS
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Column k=2 of A316622 and A316623.
Cf. A000409, A000410, A002820, A005329, A046747, A048651, A028365.
Sequence in context: A181013 A003720 A306837 * A198176 A264358 A264796
Adjacent sequences: A002881 A002882 A002883 * A002885 A002886 A002887
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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