A008964 Number of elements of order <= 2 in group of n X n upper triangular matrices over GF(2).

1, 2, 6, 28, 192, 1952, 28800, 618496, 19132416, 853508096, 54584672256, 5007197863936, 656391329021952, 122997864673574912, 32863344789678981120, 12522141669213840867328, 6792468641646224590503936, 5245768641385222636392415232

Offset

1,2

Comments

a(n) is also the number of n X n upper triangular matrices B over GF(2) such that B^2 = 0. This follows since B^2 = 0 iff (I+B)^2=I. - Hays Whitlatch, Sep 23 2025

Links

Table of n, a(n) for n=1..18.

Joshua Cooper and Hays Whitlatch, Counting Cholesky factorizations of the zero matrix over F_2, arXiv:2512.12496 [math.CO], 2025.

Shalosh B. Ekhad and Doron Zeilberger, The Number of Solutions of X^2 = 0 in Triangular Matrices Over GF(q), Elec. J. Comb. 3(1) (1996).

I. M. Isaacs and Dikran B. Karagueuzian, Involutions and characters of upper triangular matrix groups, Math. Comp. 74 (2005) 2027-2033.

Formula

From Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 30 2001: (Start)

a(2*n) = Sum_{j=-floor(n/3)..floor(n/3)} (binomial(2*n, n - 3*j) - binomial(2*n, n - 3*j - 1)) * 2^(n^2 - 3*j^2 - j).

a(2*n+1) = Sum_{j=-floor(n/3)..floor(n/3)} (binomial(2*n + 1, n - 3*j) - binomial(2*n + 1, n - 3*j - 1)) * 2^(n^2 + n - 3*j^2 - 2*j). (End)

a(n) = Sum_{r=0..floor(n/2)} f(n, r) where f(n, 0) = 1, f(1, r) = 0, f(n, r) = 2^r * f(n - 1, r) + (2^(n-r) - 2^(r-1)) * f(n - 1, r - 1). - Sean A. Irvine, Apr 15 2018

Crossrefs

Cf. A063505.

Sequence in context: A084870 A372349 A111342 * A356577 A058128 A229112

Adjacent sequences: A008961 A008962 A008963 * A008965 A008966 A008967

Keyword

nonn

Author

Dikran Karagueuzian (dikran(AT)math.wisc.edu)

Extensions

More terms from Sean A. Irvine, Apr 15 2018

Status

approved