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%I #84 Mar 03 2022 23:39:07
%S 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 48,50,52,54,56,60,62,64,66,67,69,71,73,75,77,79,81,83,85,89,90,94,96,
%U 98,102,104,106,108,110,112,114,115,117,119,123,125,127,129,131
%N 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 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 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 L. Edson Jeffery, <a href="/A232499/a232499.pdf">Illustration of the first few terms</a>.
%H Rajan Murthy, <a href="/A232499/a232499.gif">Graph of sequence</a>
%H Rajan Murthy, <a href="/A232499/a232499_1.gif">Graph of intervals</a>
%H Rajan Murthy, <a href="/A232499/a232499.png">Diagram depicting A(13)</a>
%e 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 (* 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 First differences are in A229904.
%Y 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 The radii corresponding to the terms are given by the square roots of A000404.
%Y Cf. A001481, A057961.
%Y Cf. A237707 (3-dimensional analog), A239353 (4-dimensional analog).
%K nonn
%O 1,2
%A _Rajan Murthy_ and _Vale Murthy_, Nov 24 2013
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