A036698 a(n) is the number of Gaussian integers z=a+bi satisfying |z|<=n, a>0, b>=0.

0, 1, 3, 7, 12, 20, 28, 37, 49, 63, 79, 94, 110, 132, 153, 177, 199, 225, 252, 282, 314, 343, 379, 413, 448, 490, 530, 572, 613, 657, 705, 750, 802, 852, 906, 963, 1013, 1073, 1128, 1194, 1256, 1315, 1381, 1447, 1519, 1590, 1656

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Index entries for Gaussian integers and primes

Crossrefs

Cf. A000603, A014200.

Programs

C

typedef unsigned long ulong;
ulong A036698(ulong i)
{
    const ulong ring = i*i;
    ulong result = 0;
    for(ulong a = 1; a <= i; a++)
    {
        const ulong a2 = a*a;
        for(ulong b = 0; b <= i; b++)
        {
            ulong z = a2 + b*b;
            if ( ring >= z ) result++;
        }
    }
    return result;
} /* Oskar Wieland, Apr 02 2013 */

PARI

a(n)=my(n2=n^2);sum(a=1,n,sqrtint(n2-a^2)+1) \\ Charles R Greathouse IV, Apr 03 2013

PARI

a(n) = sum(k=1, n^2, sumdiv(k, d, kronecker(-4, k/d))); \\ Seiichi Manyama, Dec 20 2021

Formula

a(n) = A000603(n) - n - 1.

a(n) = n^2 * Pi/4 + O(n). - Charles R Greathouse IV, Apr 03 2013

a(n) = A014200(n^2). - Seiichi Manyama, Dec 20 2021