A011957 Number of ZnS polytypes that repeat after n layers.

0, 1, 1, 1, 1, 2, 3, 6, 10, 18, 31, 59, 105, 198, 365, 688, 1285, 2438, 4599, 8755, 16647, 31806, 60787, 116570, 223696, 430290, 828514, 1598025, 3085465, 5965612, 11545611, 22370304, 43383539, 84216330, 163617801, 318148208, 619094385, 1205609454, 2349383925, 4581307186, 8939118925, 17452582356

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000048 (number of 3-elements orbits).

Cf. A165920 (number of 2-elements orbits).

Cf. A165921 (number of 6-elements orbits).

Programs

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);
A011957[n_] := Module[{an}, If[n <= 2, Return[n - 1]]; an =A001037[n]/6;
  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 *)

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 );
}
/* simplified version (merging the routines for A011957 and A165921 above): */
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 */

Formula

(see PARI code)

Extensions

Incorrect formula removed and terms verified by Joerg Arndt, Jul 12 2012