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

%I A165920 #28 Feb 02 2023 11:35:51
%S A165920 1,0,1,1,2,3,6,10,19,33,62,112,210,387,728,1360,2570,4845,9198,17459,
%T A165920 33288,63519,121574,232960,447392,860265,1657009,3195465,6170930,
%U A165920 11930100,23091222,44738560,86767016,168428805,327235602,636289024,1238188770,2411205111,4698767640,9162588158,17878237850
%N A165920 Number of 2-elements orbits of S3 action on irreducible polynomials of degree 3n, n > 0, over GF(2).
%C A165920 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
%H A165920 Robert Israel, <a href="/A165920/b165920.txt">Table of n, a(n) for n = 1..3331</a>
%H A165920 J. E. Iglesias, <a href="https://www.researchgate.net/publication/7166727">Enumeration of polytypes of MX and MX_2 through the use of the symmetry of the Zhadanov symbol</a>, Acta Cryst. A 62 (3) (2006) 176-194, Table 1.
%H A165920 J.-F. Michon, P. Ravache, <a href="https://dx.doi.org/10.1016/j.ffa.2010.01.004">On different families of invariant irreducible polynomials over F_2</a>, Finite fields & Applications 16 (2010) 163-174.
%F A165920 a(n) = (sum_{d|n, n/d != 0 mod 3} mu(n/d)*(2^d - (-1)^d))/(3n).
%e A165920 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.
%p A165920 f:= proc(n) local D,d;
%p A165920   D:=remove(d -> (n/3/d)::integer, numtheory:-divisors(n));
%p A165920   add(numtheory:-mobius(n/d)*(2^d - (-1)^d),d=D)/(3*n)
%p A165920 end proc:
%p A165920 map(f, [$1..100]); # _Robert Israel_, Jul 14 2019
%t A165920 a[n_] := Sum[If[Mod[n/d, 3] == 0, 0, MoebiusMu[n/d]*(2^d - (-1)^d)/(3n)], {d, Divisors[n]}];
%t A165920 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 02 2023 *)
%o A165920 (PARI)
%o A165920 L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );
%o A165920 a(n) = sum(k=0, n, if( (n+k)%3==1, L(n, k), 0 ) ) / n;
%o A165920 vector(55,n,a(n))
%o A165920 /* _Joerg Arndt_, Jun 28 2012 */
%o A165920 (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
%Y A165920 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.
%K A165920 easy,nonn
%O A165920 1,5
%A A165920 _Jean Francis Michon_, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Sep 30 2009