A055410 Number of points in Z^4 of norm <= n.

1, 9, 89, 425, 1281, 3121, 6577, 11833, 20185, 32633, 49689, 72465, 102353, 140945, 190121, 250553, 323721, 411913, 519025, 643441, 789905, 961721, 1156217, 1380729, 1638241, 1927297, 2257281, 2624417, 3035033, 3490601, 4000425

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..500 from Andrew Howroyd)

Crossrefs

Column k=4 of A302997.

Cf. A046895 (sizes of successive clusters in Z^4 lattice).

Programs

C

int A055410(int i)
{
    const int ring = i*i;
    int result = 0;
    for(int a = -i; a <= i; a++)
        for(int b = -i; b <= i; b++)
            for(int c = -i; c <= i; c++)
                for(int d = -i; d <= i; d++)
                    if ( ring >= a*a + b*b + c*c + d*d ) result++;
    return result;
} /* Oskar Wieland, Apr 08 2013 */

Mathematica

a[n_] := SeriesCoefficient[EllipticTheta[3, 0, x]^4/(1 - x), {x, 0, n^2}];
a /@ Range[0, 30] (* Jean-François Alcover, Sep 23 2019, after Ilya Gutkovskiy *)

PARI

N=66;  q='q+O('q^(N^2));
t=Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^4/(1-q)); /* A046895 */
vector(sqrtint(#t),n,t[(n-1)^2+1])
/* Joerg Arndt, Apr 08 2013 */

Python

from math import isqrt
def A055410(n): return 1+((-(s:=n**2)*(n+1)+sum((q:=s//k)*((k<<1)+q+1) for k in range(1,n+1))&-1)<<2)+(((t:=isqrt(m:=s>>2))**2*(t+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))&-1)<<4) # Chai Wah Wu, Jun 24 2024

Formula

a(n) = A046895(n^2). - Joerg Arndt, Apr 08 2013

a(n) = [x^(n^2)] theta_3(x)^4/(1 - x), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 14 2018