A272268 Records in A001481 that are more than twice the previous record.

1, 4, 9, 20, 41, 85, 173, 349, 701, 1405, 2813, 5629, 11261, 22525, 45053, 90113, 180232, 360466, 720937, 1441877, 2883761, 5767525, 11535053, 23070112, 46140228, 92280457, 184560921, 369121849, 738243700, 1476487402, 2952974809, 5905949620, 11811899241

Offset: 1

Author

M. Sinan Kul, Apr 24 2016

Keywords

• nonn

Comments

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.

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.

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.

Links

• M. Sinan Kul, PARI/GP code

Crossrefs

Cf. A001481.

Programs

Mathematica

NestList[SelectFirst[Range[2 # + 1, 5 #], SquaresR[2, #] > 0 &] &, 1, 25] (* Michael De Vlieger, Apr 25 2016, Version 10 *)