A165920 -id:A165920 - cp's OEIS frontend

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

N. J. A. Sloane

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]

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.

Cf. A000048 (number of 3-elements orbits).
Cf. A165920 (number of 2-elements orbits).
Cf. A165921 (number of 6-elements orbits).

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 *)

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 */

(see PARI code)

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

A165912 Number of alternating polynomials of degree 3n in GF(2)[X], n>0.

Original entry on oeis.org

2, 0, 2, 2, 4, 6, 12, 20, 38, 66, 124, 224, 420, 774, 1456, 2720, 5140, 9690, 18396, 34918, 66576, 127038, 243148, 465920, 894784, 1720530, 3314018, 6390930, 12341860, 23860200, 46182444, 89477120, 173534032, 336857610, 654471204, 1272578048, 2476377540, 4822410222, 9397535280

Offset: 1

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

We define an alternating polynomial as follows: let I be the set of irreducible polynomials of degree > 1 over GF(2) and Sym_3 the symmetric group on 3 elements. For a polynomial P in I of degree n, we define P*(X) = X^n P(1/X) and P+(X) = P(X+1). The operators define an action of the group Sym_3 over I. Then an alternating polynomial is defined by the property that P*=P+.

The degree of an alternating polynomial is always 0 mod 3. The numbers in the sequence are always even. These polynomials are invariant under the action of the alternating subgroup Alt_3 of S3.

G. C. Greubel, Table of n, a(n) for n = 1..3300
J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2, Finite fields & Applications 16 (2010) 163-174

A001037 is the enumeration by degree of the polynomials of the set I.

A000048 is the enumeration by degree of the polynomials such that P=P* (self-reciprocal polynomials) which is the same as the one for the polynomials such that P=P+ or P=((P+)*)+.

a[n_] := 2*DivisorSum[n, Boole[Mod[n/#, 3] != 0] MoebiusMu[n/#]*(2^# - (-1)^#) &]/(3 n); Array[a, 40] (* Jean-François Alcover, Dec 03 2015, adapted from 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!=0, L(n, k), 0 ) ) / n;
vector(55,n,a(n))
/* Joerg Arndt, Jun 28 2012 */

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

a(n) = 2 * A165920(n).

Edited by N. J. A. Sloane, May 15 2010

A165921 Number of 6-elements orbits of S3 action on irreducible polynomials of degree n > 1 over GF(2).

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 9, 15, 31, 53, 105, 189, 363, 672, 1285, 2407, 4599, 8704, 16641, 31713, 60787, 116390, 223696, 429975, 828495, 1597440, 3085465, 5964488, 11545611, 22368256, 43383477, 84212475, 163617801, 318140816, 619094385, 1205595657, 2349383715, 4581280972, 8939118925, 17452532040, 34093383807

Offset: 2

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

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

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.

J.-F. Michon, P. Ravache, On different families of invariant irreducible polynomials over F_2[X], Finite fields & Applications, 16 (2010) 163-174.

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.
Cf. A011957.
Cf. A096060 = A165921 o A000040 (on 3..oo), a subsequence of this sequence.

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);
A165921[n_] := Module[{an},
  If[n <= 2, Return[0]];
  an = A001037[n];
  If[Mod[n, 2] == 0, an -= 3*A000048[n/2]];
  If[Mod[n, 3] == 0, an -= 2*A165920[n/3]];
  an /= 6;
  Return[an]
];
Table[A165921[n], {n, 2, 50}] (* Jean-François Alcover, Dec 02 2015, adapted from Joerg Arndt's PARI script *)

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)= /* this sequence */
{
    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 );
}
/* Joerg Arndt, Jul 12 2012 */

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

(see PARI code)

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

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

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A011957 Number of ZnS polytypes that repeat after n layers.

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A165912 Number of alternating polynomials of degree 3n in GF(2)[X], n>0.

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A165921 Number of 6-elements orbits of S3 action on irreducible polynomials of degree n > 1 over GF(2).

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