A055412 Number of points in Z^6 of norm <= n.

1, 13, 485, 4197, 23793, 84769, 252673, 622573, 1395261, 2787125, 5260181, 9249417, 15637897, 25112577, 39258381, 59174749, 87380293, 125264525, 176663297, 244000537, 332379769, 444344469, 587923621, 766764301, 990981473

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500

Crossrefs

Column k=6 of A302997.

Cf. A122510.

Programs

Mathematica

t[d_, n_] := t[d, n] = t[d, n - 1] + SquaresR[d, n]; t[d_, 0] = 1;
a[n_] := t[6, n^2];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 27 2019, after R. J. Mathar *)

Python

from math import prod
from sympy import factorint
def A055412(n):
    c = 1
    for m in range(1,n**2+1):
        f = [(p,e,(0,1,0,-1)[p&3]) for p,e in factorint(m).items()]
        c += (prod((p**(e+1<<1)-a)//(p**2-a) for p, e, a in f)<<2)-prod(((k:=p**2*a)**(e+1)-1)//(k-1) for p, e, a in f)<<2
    return c # Chai Wah Wu, Jun 21 2024

Formula

a(n) = A122510(6,n^2). - R. J. Mathar, Apr 21 2010

a(n) = [x^(n^2)] theta_3(x)^6/(1 - x), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 14 2018