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This entry is motivated by the proposal A255195 by _Mats Granvik, _, who gave the first differences of this array.
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)).
E. W. Weisstein, World of Mathematics, <a href="http://mathworld.wolfram.com/GausssCircleProblem.html">Gauss's Circle Problem </a>.
T(n, m) = floor(sqrt(n^2 - m^2)) - (m-1), n >= 0, m = 0, 1, ..., floor(n/sqrt(2)).
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T)m, (n, m) = floor(sqrt(n^2 - m^2)) - (m-1), n >= 0, m = 0,1, ..., floor(n/sqrt(2)).
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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 in the first octant.
The total number of square lattice points covered 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).
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The total number of square lattice points covered in the first quadrant by a circular disk of radius R = n is therefore 2*RS(n) - (1 + floor(n/sqrt(2))) = A000603(n).
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