A192494 -id:A192494 - cp's OEIS frontend

A192493 Numerators of squared radii of circumcircles of non-degenerate triangles with integer vertex coordinates.

1, 1, 5, 25, 25, 2, 5, 25, 25, 13, 325, 169, 65, 4, 65, 17, 425, 221, 9, 289, 1105, 169, 85, 5, 325, 85, 50, 1105, 289, 25, 2125, 625, 13, 325, 425, 1625, 169, 1105, 125, 65, 29, 2465, 4225, 1885, 725, 377, 2465, 5525, 1885, 125, 8, 145, 65, 841, 17, 841, 845, 425, 2125, 221, 6409, 9425, 9, 325, 289, 145, 1105, 37, 5365, 3145, 169, 2405, 925, 85, 1369, 4625, 481, 625, 493, 2405, 10

Offset: 1

Hugo Pfoertner, Jul 10 2011

			The smallest triangle of lattice points {(0,0),(1,0),(0,1)} has circumradius R=sqrt(2)/2, i.e., R^2=1/2. Therefore a(1)=1, A192494(1)=2.

Hugo Pfoertner, Table of n, a(n) for n = 1..9089, covering range R^2 <= 100.
Hugo Pfoertner, Circles Passing through 3 Points of the Square Lattice, illustrations up to R^2=10.

Cf. A192494 (corresponding denominators), A128006, A128007.

Cf. A346993, A346994, A346995.

A128006 Numerators of rational-valued radii of circles given by 3 distinct integer points in the euclidean plane.

Original entry on oeis.org

1, 5, 5, 2, 17, 13, 5, 13, 29, 3, 37, 25, 13, 10, 17, 25, 15, 53, 65, 4, 65, 41, 29, 25, 17, 13, 41, 73, 65, 85, 5, 101, 61, 41, 26, 37, 85, 97, 65, 109, 61, 89, 325, 17, 125, 29, 53, 65, 6, 145, 85, 61, 37, 137, 25, 51, 13, 85, 205, 73, 173, 533, 20, 34, 89, 7, 197, 169, 113, 85

Offset: 1

Heinrich Ludwig, Feb 11 2007

A triangle in the Euclidean plane defines a circumcircle. There exist only certain rational-valued circumradii if the vertices of the triangle have integer coordinates. They are rendered by the pair of sequences A128006/A128007 in increasing order.

Illustration of A128006/A128007 using Plot 2.

See A128007 for denominators.

Cf. A128008, A128009, A128010, A128011, A192493, A192494.

A128007 Denominators of rational-valued radii of circles given by 3 distinct integer points in the Euclidean plane.

Original entry on oeis.org

1, 4, 3, 1, 8, 6, 2, 5, 10, 1, 12, 8, 4, 3, 5, 7, 4, 14, 17, 1, 16, 10, 7, 6, 4, 3, 9, 16, 14, 18, 1, 20, 12, 8, 5, 7, 16, 18, 12, 20, 11, 16, 58, 3, 22, 5, 9, 11, 1, 24, 14, 10, 6, 22, 4, 8, 2, 13, 31, 11, 26, 80, 3, 5, 13, 1, 28, 24, 16, 12, 26, 22, 7, 9, 4

Offset: 1

Heinrich Ludwig, Feb 11 2007

A triangle in the Euclidean plane defines a circumcircle. There exist only certain rational-valued circumradii if the vertices of the triangle have integer coordinates. The circumradii are rendered by the pair of sequences A128006/A128007 in increasing order.

See A128006 for numerators.

Cf. A128008, A128009, A128010, A128011, A192493, A192494.

A321693 Numerator of least value of the squared diameters of the enclosing circles of all strictly convex lattice n-gons with minimal area given by A070911.

Original entry on oeis.org

2, 2, 50, 8, 10, 10, 1250, 29, 40, 52, 73, 73, 82, 82, 23290, 148, 202, 226, 317, 317, 365, 452, 500, 530

Offset: 3

Hugo Pfoertner, Nov 21 2018

Without the minimal area stipulation, the result differs for some n. (See n = 12 in the examples.) - Peter Munn, Nov 17 2022

			For n = 5, the polygon with minimal area A070911(5) = 5 and enclosing circle of least diameter is
  2         D
  |       +   +
  |     +       +
  |   +           +
  1 E               C
  | +             +
  | +          +
  | +        +
  0 A + + + B
    0 ----- 1 ----- 2 ---
.
The enclosing circle passes through points A (0,0), C (2,1) and D (1,2). Its diameter is sqrt(50/9). Therefore a(5) = 50 and A322029(5) = 9.
For n = 11, a strictly convex polygon ABCDEFGHIJKA with minimal area and enclosing circle of least diameter is
    0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
  5                          J ++++++ I
  |                      +              +
  |                  +              .     +
  |             +                            +
  4         K                      .           H
  |       +                                      +
  |     +                        .                +
  |   +                                            +
  3 A                           .                   +
  | +    .                                           +
  | +           .             .                       +
  | +                 .                                +
  2 B                        O                          G
  |   +                            .                    +
  |     +                                 .             +
  |       +                                      .      +
  1         C                                           F
  |             +                                  +
  |                  +                        +
  |                      +                +
  0                          D ++++++ E
    0 ----- 1 ----- 2 ------ 3 ------ 4 ------ 5 ------ 6
.
The diameter d of the enclosing circle is determined by points A and F, with I also lying on this circle.  d^2 = 6^2 + 2^2 = 40. Therefore a(11) = 40 and A322029(11) = 1.
n = 12 is a case where the minimal area stipulation is significant. If we take the upper 6 edges in the n = 11 illustration above and rotate them about the enclosing circle's center to generate another 6 edges, we get a 12-gon with relevant squared diameter a(11) = 40 that meets all criteria except minimal area. This 12-gon's area is 26, and to meet the minimal area A070911(12)/2 = 24, the least squared diameter achievable is 52 (see illustration in the Pfoertner link). So a(12) = 52 and A322029(12) = 1. - _Peter Munn_, Nov 17 2022

Hugo Pfoertner, Illustrations of optimal polygons for n <= 26 (2018).

Cf. A070911, A192493, A192494, A322029 (corresponding denominators).

a(21)-a(26) from Hugo Pfoertner, Dec 03 2018

A322029 Denominator of least value of the squared diameters of the enclosing circles of all strictly convex lattice n-gons with minimal area given by A070911. Numerators are A321693.

Original entry on oeis.org

1, 1, 9, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 1, 169, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 3

Hugo Pfoertner, Nov 24 2018

Cf. A070911, A192493, A192494, A321693.

a(21)-a(26) from Hugo Pfoertner, Dec 03 2018

A346124 Numbers m such that no self-avoiding walk of length m + 1 on the square lattice fits into the smallest circle that can enclose a walk of length m.

Original entry on oeis.org

1, 4, 6, 8, 12, 14, 15, 16, 18, 20, 21, 23, 24, 25, 26, 27, 28, 32, 34, 36, 38, 44, 46, 48, 52, 56, 58, 60

Offset: 1

Hugo Pfoertner and Markus Sigg, Jul 30 2021

Closed walks are allowed.

			See link for illustrations of terms corresponding to diameters D < 8.5.

Hugo Pfoertner, Examples of paths of maximum length.

The squared radii of the enclosing circles are a subset of A192493/A192494.
Cf. A037245, A122224, A127399, A127400, A127401, A266549, A316194, A346993, A346994, A346995.
Cf. A346123-A346132 similar to this sequence with other sets of turning angles.

A346994 Numerators of the squared radii corresponding to circular disks covering record numbers of grid points A346993 of the square lattice.

Original entry on oeis.org

0, 1, 1, 1, 5, 25, 2, 5, 169, 65, 17, 169, 5, 25, 13, 29, 5525, 125, 17, 481, 10, 45, 205, 19721, 1189, 25, 13, 338, 29725, 697, 29, 65, 17, 1105, 3445, 18, 4453, 40885, 4625, 481, 20, 85, 12505, 2125, 200, 89, 45, 7921, 425, 89725, 93925, 2405, 2465, 10201, 98345

Offset: 1

Hugo Pfoertner, Aug 16 2021

			0/1, 1/4, 1/2, 1/1, 5/4, 25/16, 2/1, 5/2, 169/50, 65/18, 17/4, 169/36, ...
For detailed examples see A346993 and the linked pdf.

Hugo Pfoertner, Visualization of configurations with maximum number of covered grid points.

The corresponding denominators are A346995.
Cf. A346993, A123690, A346124, A346784, A346785.
All terms of a(n)/A346995(n) with the sole exception of 1/4 are terms of A192493/A192494.

A331244 Triangles with integer sides i <= j <= k sorted by radius of enclosing circle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the shortest side i. The other sides are in A331245 and A331246.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 3, 2, 3, 1, 2, 3, 4, 2, 3, 3, 1, 2, 4, 3, 4, 5, 2, 3, 3, 4, 1, 4, 2, 3, 5, 4, 5, 6, 2, 3, 3, 4, 4, 5, 4, 1, 2, 5, 3, 4, 6, 5, 6, 2, 3, 3, 4, 4, 5, 5, 4, 1, 6, 2, 5, 7, 3, 4, 6, 5, 7, 6, 7, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 1, 5, 2, 3, 7, 6, 4

Offset: 1

Hugo Pfoertner, Jan 20 2020

nonn

The enclosing circle differs from the circumcircle by limiting the radius to (longest side)/2 for obtuse triangles, i.e., those with i^2 + j^2 < k^2.

			List of triangles begins:
   n
   |     R^2
   |     |    i .... (this sequence)
   |     |    | j .. (A331245)
   |     |    | | k  (A331246)
   |     |    | | |
   1    1/ 3  1 1 1
   2   16/15  1 2 2
   3    4/ 3  2 2 2
   4    9/ 4  2 2 3  obtuse
   5   81/35  1 3 3
   6   81/32  2 3 3
   7    3/ 1  3 3 3
   8    4/ 1  2 3 4  obtuse
   9   81/20  3 3 4
  10  256/63  1 4 4
  11   64/15  2 4 4
  12  256/55  3 4 4
  13   16/ 3  4 4 4
  14   25/ 4  2 4 5  obtuse
  15   25/ 4  3 3 5  obtuse
  16   25/ 4  3 4 5
  17  625/99  1 5 5

Cf. A128006, A128007, A192493, A192494, A331227, A331228, A331251, A331252, A331253, A331254, A331255, A331256.

Cf. A331245 (middle side), A331246 (longest side).

A348469 Maximal number of squares that can be formed from the grid points in a qualifying circular region of the plane that contains exactly n points of a square grid.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 7, 8, 11, 13, 15, 17, 20, 22, 25, 28, 32, 37, 40, 43, 47, 51, 56, 60, 65, 70, 75, 81, 88, 92, 97, 103, 109, 117, 123, 130, 137, 144, 151, 158, 166, 175, 182, 189, 198, 207, 216, 226, 237, 245, 254, 263, 272, 282, 293, 303, 314

Offset: 0

Sascha Kurz and Peter Munn, Oct 19 2021

nonn

A circular region qualifies if (1) 3 (or more) grid points are incident on its circumference, or (2) it is an adjustment of a circular region, D, as defined in (1), so as to exclude only one or only consecutive grid points on the circumference of D. (Any such points on the circumference of D can be excluded by perturbing the center and radius of D by compatible but arbitrarily small amounts.)
The sequence definition is designed to help investigate the extent to which terms of A051602 can be equalled using only circular regions, while facilitating quicker calculation of terms. At the time of first submission, it is not clear to the authors that the qualification on the circular regions excludes any otherwise permissible configuration of points. In the absence of this knowledge, the qualification allows for the desired quicker calculation.
See A051602 for more information, references and links related to the general problem.

			For the following examples, we refer to _Hugo Pfoertner_'s pictorial catalog of circles passing through 3 or more grid points (see links section). Each illustration in the catalog is headed by the relevant terms of the sequences that give the squared radii of the circles, e.g. "A192493(5) = 25, A192494(5) = 16". The last line underneath each illustration gives the number of grid points in the circular region, e.g. "4+3=7" indicates 7 grid points total, of which 3 are on the circumference.
For n = 10, in the Pfoertner catalog we see the only circular region with 10 points corresponds to A192493(8). From the points in the illustration for A192493(8), 7 squares can be formed. This matches A051602(10) = 7, the maximal number of squares that can be formed from 10 points, so a(10) = 7.
For n = 20, in the Pfoertner catalog the only circular region with 20 points corresponds to A192493(29). From the points in the illustration for A192493(29), 31 squares can be formed. The circular region corresponding to A192493(24) has 21 points. From the points in the illustration for A192493(24) with any circumferential point excluded to leave 20 points, 32 squares can be formed. From a comprehensive search not detailed here, we ascertain that 32 is the most squares that can be formed from a 20 point configuration defined in the specified manner, so a(20) = 32.

Sascha Kurz, Table of n, a(n) for n = 0..100
Sascha Kurz, C++ program
Hugo Pfoertner, Circles Passing through 3 Points of the Square Lattice, illustrations up to R^2=10.

Cf. A051602, A192493/A192494.

a(n) <= A051602(n).

A355884 Number of circles in an n X n grid passing through at least three points.

Original entry on oeis.org

0, 0, 1, 34, 223, 997, 3402, 9141, 21665, 46390, 90874, 167539, 293443, 487082, 781537, 1209469, 1816528, 2661113, 3822203, 5369662, 7420495, 10086360, 13494376

Offset: 0

Sharvil Kesarwani, Jul 20 2022

C. Lin, Number of circles in configuration, Mathematics Stack Exchange, 2013.

Cf. A192493, A192494, A357301.

\\ after user joriki's Java code at Mathematics Stack Exchange link
circles(n) = {
  my(C = List());
  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);
                my( b2 = x2^2 + y2^2 - x1^2 - y1^2,
                  b3 = x3^2 + y3^2 - x1^2 - y1^2,
                  x = b2 * ay3 - b3 * ay2,
                  y = ax2 * b3 - ax3 * b2,
                  gc = gcd(gcd(x, y), den)
                );
              if (den < 0, gc = -gc);
                x /= gc; y /= gc; den /= gc;
                my( dx = x - den * x1,
                  dy = y - den * y1,
                  s = dx^2 + dy^2
                );
              listput(C, [x, y, s, den])
  ))))));
  Set(C)
};
for (k = 0, 10, print1(#circles(k), ", "))  \\ Hugo Pfoertner, Sep 22 2022

cp's OEIS Frontend

A192493 Numerators of squared radii of circumcircles of non-degenerate triangles with integer vertex coordinates.

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A128006 Numerators of rational-valued radii of circles given by 3 distinct integer points in the euclidean plane.

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A128007 Denominators of rational-valued radii of circles given by 3 distinct integer points in the Euclidean plane.

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A321693 Numerator of least value of the squared diameters of the enclosing circles of all strictly convex lattice n-gons with minimal area given by A070911.

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A322029 Denominator of least value of the squared diameters of the enclosing circles of all strictly convex lattice n-gons with minimal area given by A070911. Numerators are A321693.

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A346124 Numbers m such that no self-avoiding walk of length m + 1 on the square lattice fits into the smallest circle that can enclose a walk of length m.

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A346994 Numerators of the squared radii corresponding to circular disks covering record numbers of grid points A346993 of the square lattice.

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A331244 Triangles with integer sides i <= j <= k sorted by radius of enclosing circle, and, in case of ties, lexicographically by side lengths (smallest first). The sequence gives the shortest side i. The other sides are in A331245 and A331246.

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A348469 Maximal number of squares that can be formed from the grid points in a qualifying circular region of the plane that contains exactly n points of a square grid.

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Formula

A355884 Number of circles in an n X n grid passing through at least three points.

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