A255238 Triangle T(n, m) of numbers of points of a square lattice covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m in the first quadrant.

1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 4, 3, 1, 6, 5, 5, 5, 4, 1, 7, 6, 6, 6, 5, 4, 1, 8, 7, 7, 7, 6, 5, 4, 1, 9, 8, 8, 8, 7, 7, 6, 4, 1, 10, 9, 9, 9, 9, 8, 7, 6, 5, 1, 11, 10, 10, 10, 10, 9, 9, 8, 7, 5, 1

Offset: 0

Author

Wolfdieter Lang, Mar 12 2015

Keywords

• nonn
• easy
• tabl

Comments

This entry is motivated by the proposal A255195 by Mats Granvik.

See the MathWorld link on Gauss's circle problem.

The first quadrant of a square lattice (x, y) with x = n >= 0, y = m >= 0, is considered. The number of lattice points 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, ..., n.

The same numbers occur if x and y are interchanged.

One could also consider the row reversed triangle.

The row sums give R(n) = A000603(n), n >= 0.

The alternating row sums give A255239(n), n >= 0.

The total number of square lattice points covered by a circular disk of radius n is A000328(n) = 4*R(n) - (4*n+3).

Examples

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

Links

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

Crossrefs

Cf. A000603, A000328, A255239.

Formula

T(n, m) = 1 + floor(sqrt(n^2 - m^2)), 0 <= m <= n.