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

2,6

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

Table of n, a(n) for n=2..43.

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

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

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 *)

Prog

(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

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.

Sequence in context: A253197 A255064 A369116 * A030136 A320797 A330468

Adjacent sequences: A165918 A165919 A165920 * A165922 A165923 A165924

Keyword

easy,nonn

Author

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

Extensions

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

Status

approved