

A123290


Number of distinct C(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
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OFFSET

2,1


COMMENTS

Or, the number of commutators in a central extension of order 2^C(n+1,2) covering the elementary Abelian 2group of order 2^n. Probably also equal to the number of symmetric (n1)by(n1) matrices with entries in GF(2) of rank less than or equal to 2 and the number of skewsymmetric nbyn matrices in GF(2) of rank less than or equal to 2.


REFERENCES

LuiseCharlotte Kappe and Robert F. Morse, On Commutators in groups. Groups St. Andrews 2005. Vol. 2, 531558, London Math. Soc. Lecture Note Ser., 340, Cambridge Univ. Press, Cambridge, 2007.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 2..1000


FORMULA

a(n) = (2^(2n1)  2^n  2^(n1) + 4)/3 = 1 + (2^n  1)*(2^(n1)  1)/3
a(2)=2, a(3)=8, a(4)=36, a(n)=7*a(n1)14*a(n2)+8*a(n3) [From Harvey P. Dale, Oct 03 2011]
a(2)=2, a(n) = 4*(a(n1)1)+2^(n1).  Vincenzo Librandi, Oct 04 2011


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^n1) (2^(n1)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*n1)  2^n  2^(n1) + 4)/3: n in [2..30]]; // Vincenzo Librandi, Oct 04 2011


CROSSREFS

Sequence in context: A076122 A236626 A248861 * A228791 A088675 A228197
Adjacent sequences: A123287 A123288 A123289 * A123291 A123292 A123293


KEYWORD

nonn


AUTHOR

David Savitt (savitt(AT)math.arizona.edu), Oct 10 2006, Oct 12 2006


STATUS

approved



