A165920 Number of 2-elements orbits of S3 action on irreducible polynomials of degree 3n, n > 0, over GF(2).

1, 0, 1, 1, 2, 3, 6, 10, 19, 33, 62, 112, 210, 387, 728, 1360, 2570, 4845, 9198, 17459, 33288, 63519, 121574, 232960, 447392, 860265, 1657009, 3195465, 6170930, 11930100, 23091222, 44738560, 86767016, 168428805, 327235602, 636289024, 1238188770, 2411205111, 4698767640, 9162588158, 17878237850

Offset: 1

Author

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

Keywords

• easy
• nonn

Comments

Arndt's PARI code computes a(n) as the sum, divided by n, of every 3rd term in row n of L = A050186 = Möbius transform of binomials, starting with k = (1-n) mod 3 (nonnegative remainder), where k = 0 and k = n give L(n, k) = 0 and can be omitted. Cf. A053727, EXAMPLE and second PROGRAM. - M. F. Hasler, Sep 27 2018

Examples

			Illustrating computation via L = A050186, cf. COMMENTS: a(1) = [L(1,0)] = 0. a(2) = [L(2,2)] = 0. a(3) = L(3,1)/3 = 3/3 = 1. a(4) = ([L(4,0)] + L(4,3))/4 = 4/4 = 1. a(5) = (L(5,2) + [L(5,5)])/5 = 10/5 = 2. In [...] are terms L(n,0) = L(n,n) = 0.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..3331
• J. E. Iglesias, Enumeration of polytypes of MX and MX_2 through the use of the symmetry of the Zhadanov symbol, Acta Cryst. A 62 (3) (2006) 176-194, Table 1.
• J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174.

Crossrefs

This sequence is the half of A165912 (the number of alternate polynomials). A001037 is the enumeration by degree of the polynomials of I. A000048 is the number of 3-elements orbits of S3 action on I.

Programs

Maple

f:= proc(n) local D,d;
  D:=remove(d -> (n/3/d)::integer, numtheory:-divisors(n));
  add(numtheory:-mobius(n/d)*(2^d - (-1)^d),d=D)/(3*n)
end proc:
map(f, [$1..100]); # Robert Israel, Jul 14 2019

Mathematica

a[n_] := Sum[If[Mod[n/d, 3] == 0, 0, MoebiusMu[n/d]*(2^d - (-1)^d)/(3n)], {d, Divisors[n]}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 02 2023 *)

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==1, L(n, k), 0 ) ) / n;
vector(55,n,a(n))
/* Joerg Arndt, Jun 28 2012 */

PARI

A165920(n,k=(1-n)%3)=sum(i=0,(n-k)\3,A050186(n,k+3*i))\n \\ For illustration. - M. F. Hasler, Sep 30 2018

Formula

a(n) = (sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n).