cp's OEIS Frontend

A232499 Number of unit squares, aligned with a Cartesian grid, completely within the first quadrant of a circle centered at the origin ordered by increasing radius.

%I A232499 #84 Mar 03 2022 23:39:07
%S A232499 1,3,4,6,8,10,11,13,15,17,19,20,22,24,26,28,30,33,35,37,39,41,45,47,
%T A232499 48,50,52,54,56,60,62,64,66,67,69,71,73,75,77,79,81,83,85,89,90,94,96,
%U A232499 98,102,104,106,108,110,112,114,115,117,119,123,125,127,129,131
%N A232499 Number of unit squares, aligned with a Cartesian grid, completely within the first quadrant of a circle centered at the origin ordered by increasing radius.
%C A232499 The interval between terms reflects the number of ways a square integer can be partitioned into the sum of two square integers in an ordered pair. As examples, the increase from a(1) to a(2) from 1 to 3 is due to the inclusion of (1,2) and (2,1); and the increase from a(2) to a(3) is due to the inclusion of (2,2). Larger intervals occur when there are more combinations, such as, between a(17) and a(18) when (1,7), (7,1), and (5,5) are included.
%H A232499 Robert G. Wilson v, <a href="/A232499/b232499.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2623 from Rajan Murthy).
%H A232499 L. Edson Jeffery, <a href="/A232499/a232499.pdf">Illustration of the first few terms</a>.
%H A232499 Rajan Murthy, <a href="/A232499/a232499.gif">Graph of sequence</a>
%H A232499 Rajan Murthy, <a href="/A232499/a232499_1.gif">Graph of intervals</a>
%H A232499 Rajan Murthy, <a href="/A232499/a232499.png">Diagram depicting A(13)</a>
%e A232499 When radius of the circle exceeds 2^(1/2), one square is completely within the circle until the radius reaches 5^(1/2) when three squares are completely within the circle.
%t A232499 (* An empirical solution *) terms = 100; f[r_] := Sum[Floor[Sqrt[r^2 - n^2]], {n, 1, Floor[r]}]; Clear[g]; g[m_] := g[m] = Union[Table[f[Sqrt[s]], {s, 2, m }]][[1 ;; terms]]; g[m = dm = 4*terms]; g[m = m + dm]; While[g[m] != g[m - dm], Print[m]; m = m + dm]; A232499 = g[m]  (* _Jean-François Alcover_, Mar 06 2014 *)
%Y A232499 First differences are in A229904.
%Y A232499 The first differences must be odd at positions given in A024517 by proof by symmetry as r^2=2*n^2 is on the x=y line.
%Y A232499 The radii corresponding to the terms are given by the square roots of A000404.
%Y A232499 Cf. A001481, A057961.
%Y A232499 Cf. A237707 (3-dimensional analog), A239353 (4-dimensional analog).
%K A232499 nonn
%O A232499 1,2
%A A232499 _Rajan Murthy_ and _Vale Murthy_, Nov 24 2013