cp's OEIS Frontend

242 <= a(9) <= 244.

For odd values of n, we have 2^(n-1) - 2^((n-1)/2) <= a(n) <= 2*floor(2^(n-2) - 2^(n/2-2)).

a(n) is the size of the conjugacy class in GL(n,GF(2)) corresponding to the companion matrix of x^n - 1. It can be given by the number of n X n invertible matrices over GF(2) divided by the number of n X n circulant invertible matrices over GF(2) (i.e., the centralizer of the companion matrix of x^n - 1).

If m is odd, x^m-1 has no multiple roots so that every matrix with characteristic polynomial x^m-1 also has x^m-1 as its minimal polynomial. Hence, a(m) = A089035(m). - Geoffrey Critzer, Jul 24 2025

Equivalently, a(n) is the minimum number for which there exists a subset S of GF(2)^n with a(n) elements which spans GF(2)^n as a vector space and Sum_{s in S} f(s) = 0 for all n-bit Boolean functions of algebraic degree at most 2.

Let q be a prime power. Each function from GF(q) to itself can be uniquely represented by a polynomial in GF(q)[X] of degree at most q-1. A polynomial g in GF(q)[X] is said to be a permutation polynomial, if the function x -> g(x) is a permutation of GF(q). a(n) gives the number of permutation polynomials in GF(2^n)[X] of degree at most 2^n-1 whose coefficients are all in the prime field GF(2).

Let GF(p) be a subfield of GF(q). The permutations of GF(q) of the form x -> g(x), where g is a polynomial with coefficients in GF(p), form a subgroup of the permutations of GF(q) (see Carlitz and Hayes, 1972).