1,1

We define alternate polynomial: let I be the set of the irreducible polynomials of degree > 1 over GF(2) and S3 the symmetric group on a set of 3 elements. Now, for a polynomial P in I of degree n, we define P*(X) = X^n P(1/X) and P+(X) = P(X+1). The operators define an action of the group S3 over I. Then, an alternate polynomial is defined by the property P*=P+.

The degree of an alternate is always equal to 0 mod 3. The numbers in the sequence are always even. These polynomials are invariant under the action of the alternate subgroup A3 of S3.

J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2[X], submitted to Finite fields & Applications, 2009.

a(n) = 2*(sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n)

A001037 is the enumeration by degree of the polynomials of the set I. A000048 is the enumeration by degree of the polynomials such that P=P* (self-reciprocal polynomials) which is the same as the one for the polynomials such that P=P+ or P=((P+)*)+.

Sequence in context: A137430 A002121 A118658 this_sequence A071055 A078052 A056458

Adjacent sequences: A165909 A165910 A165911 this_sequence A165913 A165914 A165915

easy,nonn

Jean-Francis Michon, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009