OFFSET
0,2
COMMENTS
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).
LINKS
E. W. Weisstein, World of Mathematics, Gauss's Circle Problem.
FORMULA
T(n, m) = floor(sqrt(n^2 - m^2)) - (m-1), n >= 0, m = 0, 1, ..., floor(n/sqrt(2)).
EXAMPLE
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
...
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Mar 14 2015
STATUS
approved