A000603 -id:A000603 - cp's OEIS frontend

A255250 Array T(n, m) of numbers of points of a square lattice in the first octant covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m.

1, 2, 3, 1, 4, 2, 1, 5, 3, 2, 6, 4, 3, 2, 7, 5, 4, 3, 1, 8, 6, 5, 4, 2, 9, 7, 6, 5, 3, 2, 10, 8, 7, 6, 5, 3, 1, 11, 9, 8, 7, 6, 4, 3, 1, 12, 10, 9, 8, 7, 5, 4, 2, 13, 11, 10, 9, 8, 6, 5, 3, 1, 14, 12, 11, 10, 9, 8, 6, 4, 3, 1, 15, 13, 12, 11, 10, 9, 7, 6, 4, 2, 16, 14, 13, 12, 11, 10, 8, 7, 5, 4, 2

Offset: 0

Wolfdieter Lang, Mar 14 2015

The row length of this array (irregular triangle) is 1 + flpoor(n/sqrt(2)) = 1 + A049472(n) = 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 10, 11, ...
This entry is motivated by the proposal A255195 by Mats Granvik, who gave the first differences of this array.
See the MathWorld link on Gauss's circle problem.
The first octant of a square lattice (x, y) with n = x >= y = m >= 0 is considered. The number of lattice points in this octant covered by a circular disk of radius R = n around the origin having ordinate value y = m are denoted by T(n, m), for n >= 0 and m = 0, 1, ..., floor(n/sqrt(2)).
The row sums give RS(n) = A036702(n), n >= 0. This is the total number of square lattice points in the first octant covered by a circular disk of radius R = n.
The alternating row sums give A256094(n), n >= 0.
The total number of square lattice points in the first quadrant covered by a circular disk of radius R = n is therefore 2*RS(n) - (1 + floor(n/sqrt(2))) = A000603(n).

			The array (irregular triangle) T(n, m) begins:
n\m  0  1  2  3  4  5 6 7 8 9 10 ....
0:   1
1:   2
2:   3  1
3:   4  2  1
4:   5  3  2
5:   6  4  3  2
6:   7  5  4  3  1
7:   8  6  5  4  2
8:   9  7  6  5  3  2
9:  10  8  7  6  5  3 1
10: 11  9  8  7  6  4 3 1
11: 12 10  9  8  7  5 4 2
12: 13 11 10  9  8  6 5 3 1
13: 14 12 11 10  9  8 6 4 3 1
14: 15 13 12 11 10  9 7 6 4 2
15: 16 14 13 12 11 10 8 7 5 4  2
...

E. W. Weisstein, World of Mathematics, Gauss's Circle Problem.

Cf. A036702, A256094, A000328, A049472, A255195, A255238 (quadrant).

T(n, m) = floor(sqrt(n^2 - m^2)) - (m-1), n >= 0, m = 0, 1, ..., floor(n/sqrt(2)).

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

Original entry on oeis.org

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

Clark Kimberling

nonn

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

Cf. A000603, A014200.

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 */

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

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

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

A218706 Number of nonnegative integer solutions to x^2 + 2y^2 <= n^2.

Original entry on oeis.org

1, 2, 5, 9, 12, 19, 27, 33, 42, 54, 66, 77, 91, 105, 120, 138, 156, 175, 197, 218, 240, 263, 287, 314, 339, 367, 398, 430, 459, 493, 526, 556, 595, 637, 670, 709, 752, 794, 833, 878, 921, 965, 1018, 1065, 1112, 1163, 1215, 1266, 1317, 1370, 1433, 1492, 1544

Offset: 0

Jon Perry, Nov 04 2012

nonn

			There are 5 solutions for n=2, namely (0,0), (0,1), (1,1), (1,0) and (2,0), so a(2) = 5.

Cf. A000603, A218799.

for (i=0;i<50;i++) {
c=0;
for (a=0;a<=i;a++)
for (b=0;b<=i;b++)
if (Math.pow(a,2)+2*Math.pow(b,2)<=Math.pow(i,2)) c++;
document.write(c+", ");
}

nn = 50; t = Sort[Select[Flatten[Table[x^2 + 2*y^2, {x, 0, nn}, {y, 0, nn}]], # <= nn^2 &]]; Table[Count[t, ?(# <= n^2 &)], {n, 0, nn}] (* _T. D. Noe, Nov 06 2012 *)

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A255250 Array T(n, m) of numbers of points of a square lattice in the first octant covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m.

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

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A218706 Number of nonnegative integer solutions to x^2 + 2y^2 <= n^2.

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