This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A255238 #10 Feb 16 2025 08:33:25 %S A255238 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 A255238 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 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. %C A255238 This entry is motivated by the proposal A255195 by Mats Granvik. %C A255238 See the MathWorld link on Gauss's circle problem. %C A255238 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 A255238 The same numbers occur if x and y are interchanged. %C A255238 One could also consider the row reversed triangle. %C A255238 The row sums give R(n) = A000603(n), n >= 0. %C A255238 The alternating row sums give A255239(n), n >= 0. %C A255238 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 A255238 E. W. Weisstein, World of Mathematics, <a href="https://mathworld.wolfram.com/GausssCircleProblem.html">Gauss's Circle Problem </a>. %F A255238 T(n, m) = 1 + floor(sqrt(n^2 - m^2)), 0 <= m <= n. %e A255238 The triangle T(n, m) begins: %e A255238 n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 %e A255238 0: 1 %e A255238 1: 2 1 %e A255238 2: 3 2 1 %e A255238 3: 4 3 3 1 %e A255238 4: 5 4 4 3 1 %e A255238 5: 6 5 5 5 4 1 %e A255238 6: 7 6 6 6 5 4 1 %e A255238 7: 8 7 7 7 6 5 4 1 %e A255238 8: 9 8 8 8 7 7 6 4 1 %e A255238 9: 10 9 9 9 9 8 7 6 5 1 %e A255238 10: 11 10 10 10 10 9 9 8 7 5 1 %e A255238 11: 12 11 11 11 11 10 10 9 8 7 5 1 %e A255238 12: 13 12 12 12 12 11 11 10 9 8 7 5 1 %e A255238 13: 14 13 13 13 13 13 12 11 11 10 9 7 6 1 %e A255238 14: 15 14 14 14 14 14 13 13 12 11 10 9 8 6 1 %e A255238 15: 16 15 15 15 15 15 14 14 13 13 12 11 10 8 6 1 %e A255238 ... %Y A255238 Cf. A000603, A000328, A255239. %K A255238 nonn,easy,tabl %O A255238 0,2 %A A255238 _Wolfdieter Lang_, Mar 12 2015