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

Author

Victor S. Miller, Dec 29 2015

Keywords

• nonn

Comments

It follows from the form of the generating function that a(n) is asymptotic to alpha*2^n where alpha = Product_{m>=1} (1-(1/16)^m)*(1-2*(1/4)^m)/((1-2*(1/16)^m)*(1-(1/4)^m)). [corrected by Jason Yuen, May 19 2025]

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

Crossrefs

Cf. A006950.

Programs

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 *)

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))).

Extensions

More terms from Alois P. Heinz, Dec 29 2015