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A123290
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Number of distinct binomial(n,2)-tuples of zeros and ones that are obtained as the collection of all 2 X 2 minor determinants of a 2 X n matrix over GF(2).
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1
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2, 8, 36, 156, 652, 2668, 10796, 43436, 174252, 698028, 2794156, 11180716, 44731052, 178940588, 715795116, 2863245996, 11453115052, 45812722348, 183251413676, 733006703276, 2932028910252, 11728119835308, 46912487729836
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OFFSET
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2,1
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COMMENTS
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Or, the number of commutators in a central extension of order 2^binomial(n+1,2) covering the elementary Abelian 2-group of order 2^n. Probably also equal to the number of symmetric (n-1) X (n-1) matrices with entries in GF(2) of rank less than or equal to 2 and the number of skew-symmetric n X n matrices in GF(2) of rank less than or equal to 2.
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REFERENCES
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Luise-Charlotte Kappe and Robert F. Morse, On Commutators in groups. Groups St. Andrews 2005. Vol. 2, 531-558, London Math. Soc. Lecture Note Ser., 340, Cambridge Univ. Press, Cambridge, 2007.
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LINKS
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FORMULA
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a(n) = (2^(2n-1) - 2^n - 2^(n-1) + 4)/3 = 1 + (2^n - 1)*(2^(n-1) - 1)/3
a(2)=2, a(3)=8, a(4)=36, a(n)=7*a(n-1)-14*a(n-2)+8*a(n-3). - Harvey P. Dale, Oct 03 2011
G.f.: 2*x^2*(1 - 3*x + 4*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)). - Colin Barker, Jan 26 2018
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EXAMPLE
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a(4) = 36. Let G be a central extension of order 2^C(5,2) covering (Z/2Z)^4; the commutator subgroup of G has order 2^C(4,2) = 64, so it is not the case that every element of the commutator subgroup of G is actually a commutator.
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MATHEMATICA
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Table[1+(2^n-1) (2^(n-1)-1)/3, {n, 2, 30}] (* or *) LinearRecurrence[ {7, -14, 8}, {2, 8, 36}, 30] (* Harvey P. Dale, Oct 03 2011 *)
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PROG
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(Magma) minors := function(n) F := GF(2); V := VectorSpace(F, 2*n); S := { } ; for v in V do M := Matrix(F, 2, n, ElementToSequence(v)); seq := Minors(M, 2); S := Include(S, seq); end for; return #S; end function;
(Magma) [(2^(2*n-1) - 2^n - 2^(n-1) + 4)/3: n in [2..30]]; // Vincenzo Librandi, Oct 04 2011
(PARI) Vec(2*x^2*(1 - 3*x + 4*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 26 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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David Savitt (savitt(AT)math.arizona.edu), Oct 10 2006, Oct 12 2006
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STATUS
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approved
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