A358649 Number of convergent n X n matrices over GF(2).

1, 2, 11, 205, 14137, 3755249, 3916674017, 16190352314305, 266479066904477569, 17503939768635307654913, 4593798697440979773283368449, 4819699338906053452395454422580225, 20221058158328101246044232181365184919553

Offset

0,2

Comments

A matrix A over a finite field is convergent if A^j=A^(j+1) for some j>=1. Every convergent matrix converges to an idempotent matrix. Every idempotent matrix is convergent to itself. Every nilpotent matrix is convergent to the zero matrix.

Links

Table of n, a(n) for n=0..12.

Formula

a(n) = Sum_{k=0..n} A296548(n,k)*A053763(n-k).

Sum_{n>=0} a(n)*x^n/B(n) = f(x)*e(x) where f(x)=Sum_{n>=0} q^(n^2-n)*x^n/B(n), e(x)=Sum_{n>=0} x^n/B(n), B(n)=Product_{i=0..n-1} (q^n-q^i)/(q-1)^n, and q=2. - Geoffrey Critzer, Jan 02 2025

Mathematica

nn = 12; q = 2; g[n_] := Product[q^n - q^i, {i, 0, n - 1}]; Table[Sum[g[n]/(g[k] g[n - k]) q^((n - k) (n - k - 1)), {k, 0, n}], {n, 0, nn}]

Crossrefs

Cf. A296548, A053763, A132186, A379778.

Sequence in context: A271429 A051663 A348859 * A188203 A070256 A356523

Adjacent sequences: A358646 A358647 A358648 * A358650 A358651 A358652

Keyword

nonn

Author

Geoffrey Critzer, Nov 26 2022

Status

approved