OFFSET
1,1
COMMENTS
We define an alternating polynomial as follows: let I be the set of irreducible polynomials of degree > 1 over GF(2) and Sym_3 the symmetric group on 3 elements. 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 Sym_3 over I. Then an alternating polynomial is defined by the property that P*=P+.
The degree of an alternating polynomial is always 0 mod 3. The numbers in the sequence are always even. These polynomials are invariant under the action of the alternating subgroup Alt_3 of S3.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..3300
J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174
FORMULA
a(n) = 2*(sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n).
a(n) = 2 * A165920(n).
MATHEMATICA
a[n_] := 2*DivisorSum[n, Boole[Mod[n/#, 3] != 0] MoebiusMu[n/#]*(2^# - (-1)^#) &]/(3 n); Array[a, 40] (* Jean-François Alcover, Dec 03 2015, adapted from PARI *)
PROG
(PARI) L(n, k) = sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%3!=0, L(n, k), 0 ) ) / n;
vector(55, n, a(n))
/* Joerg Arndt, Jun 28 2012 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jean Francis Michon and Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009
EXTENSIONS
Edited by N. J. A. Sloane, May 15 2010
STATUS
approved