A192494 -id:A192494 - cp's OEIS frontend

A357301 a(n) is the number of distinct radii of circles passing through at least three points in a square grid of n X n points.

0, 1, 7, 19, 48, 112, 212, 383, 641, 988, 1523, 2250, 3103, 4364, 5831, 7696, 9985, 12945, 16164, 20246, 24946, 30145, 36385, 43839, 51752, 61610, 72475, 84273, 97231, 112733

Offset: 1

Hugo Pfoertner, Sep 23 2022, following a suggestion by Ed Pegg Jr, Jan 09 2013

Proposed by Ed Pegg Jr in Mathematics StackExchange question 273348, where terms up to 212 were given, see link.

			a(2) = 1: only one possible circle with squared radius 1/2;
a(3) = 7: squared radii of possible circles are {1, 2, 1/2, 5/2, 5/4, 25/16, 25/18}.

Mathematics StackExchange, Number of circles in configuration, comment by Ed Pegg, Jan 09, 2013.

Cf. A192493, A192494, A355884.

\\ Function r2 determined using Mathematica
\\ (ArcLength [CircleThrough[{{x1, y1}, {x2, y2}, {x3, y3}}]]/(2*Pi))^2
radii(n) = {my (R=List(), r2(x1,y1,x2,y2,x3,y3) = ((x3 + (x2^2*y1 - x3^2*y1 - x1^2*y2 + x3^2*y2 - y1^2*y2 + y1*y2^2 + x1^2*y3 - x2^2*y3 + y1^2*y3 - y2^2*y3 - y1*y3^2 + y2*y3^2)/(2*(-x2*y1 + x3*y1 + x1*y2 - x3*y2 - x1*y3 + x2*y3)))^2 + (y3 - (-2*(x2 - x3)*(x1^2 - x3^2 + y1^2 - y3^2) + 2*(x1 - x3)*(x2^2 - x3^2 + y2^2 - y3^2))/(-4*x2*y1 + 4*x3*y1 + 4*x1*y2 - 4*x3*y2 - 4*x1*y3 + 4*x2*y3))^2)); for(x1=1, n, for(y1=1, n, for(x2=1, x1, for(y2=1, n, for(x3=1, x2, for(y3=1, n,
my (ax2=2*(x2-x1), ay2=2*(y2-y1), ax3=2*(x3-x1), ay3=2*(y3-y1), den=ax2*ay3 -ax3*ay2); if (den==0, next); listput (R, r2(x1,y1,x2,y2,x3,y3)))))))); Set(R)};
for (k=1,15, print1(#radii(k),", "))

A387045 Positive numbers k with property that the largest circle on the xy-plane enclosing exactly k lattice points in its interior does not exist.

Original entry on oeis.org

5, 6, 17, 18, 33, 34, 35, 36, 38, 50, 53, 54, 63, 70, 71, 72, 73, 89, 90, 97, 98, 102, 109, 110, 125, 126, 127, 128, 129, 150, 151, 165, 167, 168, 178, 188, 198, 209, 210, 217, 218, 219, 220, 221, 222, 242, 243, 257, 258, 259, 260, 277, 278, 285, 286, 294

Offset: 1

Jianqiang Zhao, Aug 14 2025

Conjecture: This sequence is infinite.

			It can be proved that the largest circle enclosing exactly 5 or 6 lattice points in the interior on the xy-plane does not exist. Number 5 is the smallest nonnegative integer having this property and 6 is the next. Therefore, a(1)=5 and a(2)=6.
Here is a brief argument. For details, please see my arxiv paper 2505.06234.
First, let C be the circle going through (-1,0) centered at (1/2,1/2). It passes exactly 8 lattice points and encloses exactly 4. Now with (-1,0) fixed on the circle we can shrink C by an infinitesimal amount to circle C' so that C' only goes through one lattice point (-1,0). Then another infinitesimal perturbation will move C' to include exactly 5 lattice points in its interior. Another infinitesimal perturbation will move C' to include exactly 6 lattice points in its interior. Therefore, if the largest circle enclosing exactly 5 or 6 interior lattice points exists, then its radius is at least sqrt(10)/2.
Second, a geometric argument shows that if the radius of a circle is at least sqrt(10)/2 then it encloses either exactly 4 interior lattice points or at least 7 interior lattice points.

Jianqiang Zhao, Table of n, a(n) for n = 1..189
Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, arXiv:2505.06234 [math.GM], 2025. This is an expanded version of the paper by Jianqiang Zhao, The Largest Circle Enclosing n Lattice Points, Geometry, Vol. 2 (2025), 12.

Cf. A387044, complement of A387045; A192493, A192494, A128006, A128007.

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A357301 a(n) is the number of distinct radii of circles passing through at least three points in a square grid of n X n points.

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