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 A123290 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). 1
 2, 8, 36, 156, 652, 2668, 10796, 43436, 174252, 698028, 2794156, 11180716, 44731052, 178940588, 715795116, 2863245996, 11453115052, 45812722348, 183251413676, 733006703276, 2932028910252, 11728119835308, 46912487729836 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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. REFERENCES 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 2..1000 Index entries for linear recurrences with constant coefficients, signature (7,-14,8). FORMULA 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 a(2)=2, a(n) = 4*(a(n-1)-1)+2^(n-1). - Vincenzo Librandi, Oct 04 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 EXAMPLE 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. MATHEMATICA 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 *) PROG (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 CROSSREFS Sequence in context: A323677 A295501 A283847 * A321110 A228791 A088675 Adjacent sequences: A123287 A123288 A123289 * A123291 A123292 A123293 KEYWORD nonn,easy AUTHOR David Savitt (savitt(AT)math.arizona.edu), Oct 10 2006, Oct 12 2006 STATUS approved

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Last modified October 3 22:39 EDT 2023. Contains 365872 sequences. (Running on oeis4.)