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 A255250 #15 Feb 16 2025 08:33:25 %S A255250 1,2,3,1,4,2,1,5,3,2,6,4,3,2,7,5,4,3,1,8,6,5,4,2,9,7,6,5,3,2,10,8,7,6, %T A255250 5,3,1,11,9,8,7,6,4,3,1,12,10,9,8,7,5,4,2,13,11,10,9,8,6,5,3,1,14,12, %U A255250 11,10,9,8,6,4,3,1,15,13,12,11,10,9,7,6,4,2,16,14,13,12,11,10,8,7,5,4,2 %N A255250 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. %C A255250 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, ... %C A255250 This entry is motivated by the proposal A255195 by _Mats Granvik_, who gave the first differences of this array. %C A255250 See the MathWorld link on Gauss's circle problem. %C A255250 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)). %C A255250 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. %C A255250 The alternating row sums give A256094(n), n >= 0. %C A255250 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). %H A255250 E. W. Weisstein, World of Mathematics, <a href="https://mathworld.wolfram.com/GausssCircleProblem.html">Gauss's Circle Problem</a>. %F A255250 T(n, m) = floor(sqrt(n^2 - m^2)) - (m-1), n >= 0, m = 0, 1, ..., floor(n/sqrt(2)). %e A255250 The array (irregular triangle) T(n, m) begins: %e A255250 n\m 0 1 2 3 4 5 6 7 8 9 10 .... %e A255250 0: 1 %e A255250 1: 2 %e A255250 2: 3 1 %e A255250 3: 4 2 1 %e A255250 4: 5 3 2 %e A255250 5: 6 4 3 2 %e A255250 6: 7 5 4 3 1 %e A255250 7: 8 6 5 4 2 %e A255250 8: 9 7 6 5 3 2 %e A255250 9: 10 8 7 6 5 3 1 %e A255250 10: 11 9 8 7 6 4 3 1 %e A255250 11: 12 10 9 8 7 5 4 2 %e A255250 12: 13 11 10 9 8 6 5 3 1 %e A255250 13: 14 12 11 10 9 8 6 4 3 1 %e A255250 14: 15 13 12 11 10 9 7 6 4 2 %e A255250 15: 16 14 13 12 11 10 8 7 5 4 2 %e A255250 ... %Y A255250 Cf. A036702, A256094, A000328, A049472, A255195, A255238 (quadrant). %K A255250 nonn,easy,tabf %O A255250 0,2 %A A255250 _Wolfdieter Lang_, Mar 14 2015