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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

LINKS

Table of n, a(n) for n=0..65.

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

FORMULA

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

EXAMPLE

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

...

CROSSREFS

Cf. A000603, A000328, A255239.

Sequence in context: A133334 A003603 A278118 * A212536 A188277 A135227

Adjacent sequences:  A255235 A255236 A255237 * A255239 A255240 A255241

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Mar 12 2015

STATUS

approved

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Last modified October 20 17:48 EDT 2018. Contains 316396 sequences. (Running on oeis4.)