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%I #14 Apr 25 2016 10:06:56
%S 1,4,9,20,41,85,173,349,701,1405,2813,5629,11261,22525,45053,90113,
%T 180232,360466,720937,1441877,2883761,5767525,11535053,23070112,
%U 46140228,92280457,184560921,369121849,738243700,1476487402,2952974809,5905949620,11811899241
%N Records in A001481 that are more than twice the previous record.
%C This list can be seen as the r^2 (square of the radius r) of the homocentric circles that are centered at the origin and pass through at least 4 lattice points, the innermost circle being the unit circle.
%C If we start with the unit circle (x^2 + y^2 = 1), the smallest circle that pass through at least four lattice points would be x^2 + y^2 = 4 with (2,0), (0,2), (-2,0), (0,-2). Similarly next circle would be x^2 + y^2 = 9 passing through (3,0), (0,3), (-3,0), (0,-3), and the next x^2 + y^2 = 20 passing through (2,4), (4,2), (-2,4), (-4,2), (-2,-4), (-4,-2), (2,-4), (4,-2), etc.
%C It is also worth mentioning that a square can be drawn with vertexes on the lattice points of a circle and the sides of that square wouldn't touch the smaller circle.
%H M. Sinan Kul, <a href="/A272268/a272268.gp.txt">PARI/GP code</a>
%t NestList[SelectFirst[Range[2 # + 1, 5 #], SquaresR[2, #] > 0 &] &, 1, 25] (* _Michael De Vlieger_, Apr 25 2016, Version 10 *)
%Y Cf. A001481.
%K nonn
%O 1,2
%A _M. Sinan Kul_, Apr 24 2016
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