A346082 Number of cyclic n X n matrices over GF(2).

1, 2, 14, 412, 50832, 25517184, 51759986688, 422000664182784, 13794938575436906496, 1805965390215106718072832, 946278871976706458877777936384, 1983897413727786229545246093886881792, 16639646499680599124923569106989157705580544, 558292116984541859085729903695019486031085083557888

Offset: 0

Author

Geoffrey Critzer, Jul 04 2021

Keywords

• nonn

Comments

An n X n matrix A is cyclic if there is a vector v in GF(2)^n such that {A^i(v) : i>=0} spans GF(2)^n. Equivalently if the characteristic polynomial of A is equal to the minimal polynomial.

Links

• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Crossrefs

Cf. A001037, A002884.

Main diagonal of A347010.

Programs

Mathematica

nn = 13; A001037 = Table[1/n Sum[MoebiusMu[n/d] 2^d, {d, Divisors[n]}], {n, 1, nn}]; Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
  Series[Product[(1 + 2^i x^i/((2^i - 1) (2^i - x^i)))^ A001037[[i]], {i, 1, nn}], {x, 0, nn}], x]

Formula

Sum_{n>=0} a(n) x^n/A002884(n) = Product_{i>=1} (1 + x^i/((2^i-1)(1-x/2)^i))^A001037(i).