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%I #9 Mar 13 2015 15:21:38
%S 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,
%T 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
%N 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.
%C This entry is motivated by the proposal A255195 by Mats Granvik.
%C See the MathWorld link on Gauss's circle problem.
%C 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.
%C The same numbers occur if x and y are interchanged.
%C One could also consider the row reversed triangle.
%C The row sums give R(n) = A000603(n), n >= 0.
%C The alternating row sums give A255239(n), n >= 0.
%C The total number of square lattice points covered by a circular disk of radius n is A000328(n) = 4*R(n) - (4*n+3).
%H E. W. Weisstein, World of Mathematics, <a href="http://mathworld.wolfram.com/GausssCircleProblem.html">Gauss's Circle Problem </a>.
%F T(n, m) = 1 + floor(sqrt(n^2 - m^2)), 0 <= m <= n.
%e The triangle T(n, m) begins:
%e n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e 0: 1
%e 1: 2 1
%e 2: 3 2 1
%e 3: 4 3 3 1
%e 4: 5 4 4 3 1
%e 5: 6 5 5 5 4 1
%e 6: 7 6 6 6 5 4 1
%e 7: 8 7 7 7 6 5 4 1
%e 8: 9 8 8 8 7 7 6 4 1
%e 9: 10 9 9 9 9 8 7 6 5 1
%e 10: 11 10 10 10 10 9 9 8 7 5 1
%e 11: 12 11 11 11 11 10 10 9 8 7 5 1
%e 12: 13 12 12 12 12 11 11 10 9 8 7 5 1
%e 13: 14 13 13 13 13 13 12 11 11 10 9 7 6 1
%e 14: 15 14 14 14 14 14 13 13 12 11 10 9 8 6 1
%e 15: 16 15 15 15 15 15 14 14 13 13 12 11 10 8 6 1
%e ...
%Y Cf. A000603, A000328, A255239.
%K nonn,easy,tabl
%O 0,2
%A _Wolfdieter Lang_, Mar 12 2015
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