A165921 - OEIS

%I #35 Oct 28 2018 17:08:58

%S 0,0,0,1,1,3,4,9,15,31,53,105,189,363,672,1285,2407,4599,8704,16641,

%T 31713,60787,116390,223696,429975,828495,1597440,3085465,5964488,

%U 11545611,22368256,43383477,84212475,163617801,318140816,619094385,1205595657,2349383715,4581280972,8939118925,17452532040,34093383807

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

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

%D 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.

%H J.-F. Michon, P. Ravache, <a href="http://dx.doi.org/10.1016/j.ffa.2010.01.004">On different families of invariant irreducible polynomials over F_2[X]</a>, Finite fields & Applications, 16 (2010) 163-174.

%F (see PARI code)

%F 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

%t L[n_, k_] := DivisorSum[GCD[n, k], MoebiusMu[#]*Binomial[n/#, k/#] &];

%t A165920[n_] := Sum[If[(n + k) ~Mod~ 3 == 1, L[n, k], 0], {k, 0, n}]/n;

%t A001037[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[#]*2^(n/#) &]/n];

%t A000048[n_] := DivisorSum[n, Mod[#, 2]*(MoebiusMu[#]*2^(n/#)) &]/(2*n);

%t A165921[n_] := Module[{an},

%t   If[n <= 2, Return[0]];

%t   an = A001037[n];

%t   If[Mod[n, 2] == 0, an -= 3*A000048[n/2]];

%t   If[Mod[n, 3] == 0, an -= 2*A165920[n/3]];

%t   an /= 6;

%t   Return[an]

%t ];

%t Table[A165921[n], {n, 2, 50}] (* _Jean-François Alcover_, Dec 02 2015, adapted from _Joerg Arndt_'s PARI script *)

%o (PARI)

%o L(n, k)=sumdiv(gcd(n, k), d, moebius(d) * binomial(n/d, k/d) );

%o A165920(n)=sum(k=0, n, if( (n+k)%3==1, L(n, k), 0 ) ) / n;

%o A001037(n)=if(n<1, n==0, sumdiv(n, d, moebius(d)*2^(n/d))/n);

%o A000048(n)=sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d)))/(2*n);

%o A165921(n)= /* this sequence */

%o {

%o     my(an);

%o     if ( n<=2, return(0) );

%o     an = A001037(n);

%o     if (n%2==0, an -= 3*A000048(n/2) );

%o     if (n%3==0, an -= 2*A165920(n/3) );

%o     an /= 6;

%o     return( an );

%o }

%o /* _Joerg Arndt_, Jul 12 2012 */

%o (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

%Y 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.

%Y Cf. A011957.

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

%K easy,nonn

%O 2,6

%A _Jean Francis Michon_, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009

%E Incorrect formula removed and more terms added by _Joerg Arndt_, Jul 12 2012