A165921 Number of 6-elements orbits of S3 action on irreducible polynomials of degree n > 1 over GF(2).

0, 0, 0, 1, 1, 3, 4, 9, 15, 31, 53, 105, 189, 363, 672, 1285, 2407, 4599, 8704, 16641, 31713, 60787, 116390, 223696, 429975, 828495, 1597440, 3085465, 5964488, 11545611, 22368256, 43383477, 84212475, 163617801, 318140816, 619094385, 1205595657, 2349383715, 4581280972, 8939118925, 17452532040, 34093383807

Offset: 2

Author

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

Keywords

• easy
• nonn

Comments

The terms are denoted h_6 in the Michon/Ravache reference.

References

• J. E. Iglesias, Enumeration of polytypes MX and MX_2 through the use of the symmetry of the Zhadanov symbol, Acta Cryst. A 62 (3) (2006) 178-194, Table 1.

Links

• J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2[X], Finite fields & Applications, 16 (2010) 163-174.

Crossrefs

A001037 is the enumeration by degree of the irreducible polynomials over GF(2), A000048 is the number of 3-elements orbits, A165920 is the number of 2-elements orbits.

Cf. A011957.

Cf. A096060 = A165921 o A000040 (on 3..oo), a subsequence of this sequence.

Programs

Mathematica

L[n_, k_] := DivisorSum[GCD[n, k], MoebiusMu[#]*Binomial[n/#, k/#] &];
A165920[n_] := Sum[If[(n + k) ~Mod~ 3 == 1, L[n, k], 0], {k, 0, n}]/n;
A001037[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[#]*2^(n/#) &]/n];
A000048[n_] := DivisorSum[n, Mod[#, 2]*(MoebiusMu[#]*2^(n/#)) &]/(2*n);
A165921[n_] := Module[{an},
  If[n <= 2, Return[0]];
  an = A001037[n];
  If[Mod[n, 2] == 0, an -= 3*A000048[n/2]];
  If[Mod[n, 3] == 0, an -= 2*A165920[n/3]];
  an /= 6;
  Return[an]
];
Table[A165921[n], {n, 2, 50}] (* Jean-François Alcover, Dec 02 2015, adapted from Joerg Arndt's PARI script *)

PARI

L(n, k)=sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d) );
A165920(n)=sum(k=0, n, if( (n+k)%3==1, L(n, k), 0 ) ) / n;
A001037(n)=if(n<1, n==0, sumdiv(n, d, moebius(d)*2^(n/d))/n);
A000048(n)=sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d)))/(2*n);
A165921(n)= /* this sequence */
{
    my(an);
    if ( n<=2, return(0) );
    an = A001037(n);
    if (n%2==0, an -= 3*A000048(n/2) );
    if (n%3==0, an -= 2*A165920(n/3) );
    an /= 6;
    return( an );
}
/* Joerg Arndt, Jul 12 2012 */

PARI

A165921(n)=if(n>2,A001037(n)-if(!bittest(n,0),3*A000048(n\2))-if(n%3==0,2*A165920(n\3)))\6 \\ Based on Joerg Arndt's code from Jul 12 2012. Take up-to-date code for other sequences from the respective record. - M. F. Hasler, Sep 27 2018

Formula

(see PARI code)

a(p) = (2^(p-1)-1)/3p = A096060(n) for all primes p = prime(n) >= 5, n >= 3: A165921 o A000040 = A096060 on the domain [3..oo) of that sequence. - M. F. Hasler, Sep 27 2018

Extensions

Incorrect formula removed and more terms added by Joerg Arndt, Jul 12 2012