A266462 The number of conjugacy classes of invertible n X n matrices over GF(2) which are squares of other such matrices.

1, 1, 2, 5, 10, 20, 41, 82, 166, 334, 667, 1336, 2682, 5360, 10724, 21467, 42936, 85876, 171786, 343574, 687184, 1374427, 2748852, 5497766, 10995706, 21991402, 43982908, 87966150, 175932383, 351864964, 703730584, 1407461288, 2814923196, 5629847656, 11259695532

Offset

0,3

Comments

It follows from the form of the generating function that a(n) is asymptotic to alpha*2^n where alpha = Product_{n>=1} (1-(1/16)^n)(1-2(1/2)^n)/ ((1-2(1/16)^n)(1-(1/4)^n))).

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..60 [Based on the table in Miller (2016)]

Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.

Index entries for matrices, binary, which are squares

Formula

G.f.: Product_{n>=1} (1-2*x^(2*n))*(1-x^(2*n))/((1-2*x^n)*(1-2*x^(4*n))(1+x^(2*n-1))).

Mathematica

terms = 35; CoefficientList[Product[(1-2x^(2n))(1-x^(2n))/((1-2x^n) (1-2x^(4n))(1+x^(2n-1))), {n, 1, terms}] + O[x]^terms, x] (* Jean-François Alcover, Aug 06 2018 *)

Crossrefs

Cf. A006950.

Sequence in context: A049938 A002460 A266613 * A293319 A267589 A006836

Adjacent sequences: A266459 A266460 A266461 * A266463 A266464 A266465

Keyword

nonn

Author

Victor S. Miller, Dec 29 2015

Extensions

More terms from Alois P. Heinz, Dec 29 2015

Status

approved