OFFSET
1,6
COMMENTS
Also the number of orbits of the symmetric group S3 action on irreducible polynomials of degree n>1 over GF(2). [Jean Francis Michon, Philippe Ravache (philippe.ravache(AT)univ-rouen.fr), Oct 04 2009]
LINKS
Dennis S. Bernstein, Omran Kouba, Counting Colorful Necklaces and Bracelets in Three Colors, arXiv:1901.10703 [math.CO], 2019.
Dennis S. Bernstein, Omran Kouba, Counting colorful necklaces and bracelets in three colors, Aequat. Math. (2019) 93: 1183.
T. J. McLarnan, The numbers of polytypes in close-packings and related structures, Zeitschrift für Kristallographie. Volume 155, Issue 3-4, Pages 269-291, ISSN (Online) 0044-2968, ISSN (Print) 1433-7266, DOI: 10.1524/zkri.1981.155.3-4.269, August 2010.
J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174.
FORMULA
(see PARI code)
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);
If[n ~Mod~ 2 == 0, an += 1/2*A000048[n/2]];
If[n ~Mod~ 3 == 0, an += 2/3*A165920[n/3]];
Return[an]
];
Table[A011957[n], {n, 1, 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)=
{
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 );
}
A011957(n)=
{
my(an);
an = A165921(n);
if (n%2==0, an += A000048(n/2) );
if (n%3==0, an += A165920(n/3) );
return( an );
}
A011957(n)=
{
my(an);
if ( n<=2, return(n-1) );
an = A001037(n) / 6;
if (n%2==0, an += 1/2 * A000048(n/2) );
if (n%3==0, an += 2/3 * A165920(n/3) );
return( an );
}
/* Joerg Arndt, Jul 12 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Incorrect formula removed and terms verified by Joerg Arndt, Jul 12 2012
STATUS
approved