|
|
A165921
|
|
Number of 6-elements orbits of S3 action on irreducible polynomials of degree n > 1 over GF(2).
|
|
2
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,6
|
|
COMMENTS
|
The terms are denoted h_6 in the Michon/Ravache reference.
|
|
REFERENCES
|
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.
|
|
LINKS
|
|
|
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 <= 2, Return[0]];
If[Mod[n, 2] == 0, an -= 3*A000048[n/2]];
If[Mod[n, 3] == 0, an -= 2*A165920[n/3]];
an /= 6;
Return[an]
];
|
|
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);
{
my(an);
if ( n<=2, return(0) );
if (n%2==0, an -= 3*A000048(n/2) );
if (n%3==0, an -= 2*A165920(n/3) );
an /= 6;
return( an );
}
|
|
CROSSREFS
|
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.
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Incorrect formula removed and more terms added by Joerg Arndt, Jul 12 2012
|
|
STATUS
|
approved
|
|
|
|