A379778 -id:A379778 - cp's OEIS frontend

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

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

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Geoffrey Critzer, Nov 26 2022

nonn

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.

Cf. A296548, A053763, A132186, A379778.

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}]

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

cp's OEIS Frontend

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

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