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A165921 Number of 6-elements orbits of S3 action on irreducible polynomials of degree n > 1 over GF(2). 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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A216075 A253197 A255064 * 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

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Last modified October 28 16:09 EDT 2021. Contains 348329 sequences. (Running on oeis4.)