cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A359931 Number of distinct circles that can be constructed from an n x n square grid of points using only a compass.

Original entry on oeis.org

8, 38, 120, 277, 544, 969, 1616, 2494, 3724, 5331, 7408, 10038, 13284, 17181, 21968, 27653, 34320, 42110, 51148
Offset: 2

Views

Author

Scott R. Shannon, Jan 19 2023

Keywords

Comments

A circle is constructed for every pair of the n x n points, the first point defines the circle's center while the second the radius distance.
No formula for a(n) is known.
See A359932 and A359933 for images of the resulting vertices and regions.

Crossrefs

Cf. A359932 (vertices), A359933 (regions), A359934 (edges), A359935 (k-gons).

A353782 Number of regions among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes using only a compass.

Original entry on oeis.org

112, 1264, 5548, 14976, 37092, 77096, 143560, 237504
Offset: 1

Views

Author

Scott R. Shannon, Mar 13 2023

Keywords

Comments

A circle is constructed for every pair of the 1 + 4n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the points is A361622(n).
No formula for a(n) is currently known.

Links

Crossrefs

Cf. A354605 (vertices), A356358 (edges), A361623 (k-gons), A361622 (distinct circles), A359933, A359860, A359253, A359570, A359046.

Formula

a(n) = A356358 - A354605(n) + 1 by Euler's formula.

A360352 Number of regions among all distinct circles that can be constructed from an n X n square grid of points when each pair of points is connected by a circle and the points lie at the ends of a diameter of the circle.

Original entry on oeis.org

12, 168, 1536, 8904, 36880, 123468, 358036, 912776, 2105976
Offset: 2

Views

Author

Scott R. Shannon, Feb 04 2023

Keywords

Comments

A circle is constructed for every pair of points on the n X n grid, the points lying at the ends of a diameter of the circle. The number of distinct circles constructed from the n X n grid is A360350(n).

Links

  • Scott R. Shannon, Image for n = 2. In this and other images the n X n grid points are shown as white dots.
  • Scott R. Shannon, Image for n = 3.
  • Scott R. Shannon, Image for n = 4.
  • Scott R. Shannon, Image for n = 5.
  • Scott R. Shannon, Image for n = 6.
  • N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)

Crossrefs

Cf. A360351 (vertices), A360353 (edges), A360354 (k-gons), A360350 (distinct circles), A359933.

Formula

a(n) = A360353(n) - A360351(n) + 1 by Euler's formula.

A359932 Number of vertices among all distinct circles that can be constructed from an n x n square grid of points using only a compass.

Original entry on oeis.org

40, 689, 7240, 38729, 151584, 488741
Offset: 2

Views

Author

Scott R. Shannon, Jan 19 2023

Keywords

Comments

A circle is constructed for every pair of the n x n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the n x n points is A359931(n).
No formula for a(n) is known.

Links

Crossrefs

Cf. A359933 (regions), A359934 (edges), A359935 (k-gons), A359931 (distinct circles), A359859, A359252.

Formula

a(n) = A359934(n) - A359933(n) + 1 by Euler's formula.

A359935 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, among all distinct circles that can be constructed from an n x n square grid of points using only a compass.

Original entry on oeis.org

0, 16, 30, 0, 412, 341, 60, 20, 4, 0, 3464, 3534, 928, 212, 48, 12, 0, 16936, 19861, 5252, 1056, 88, 52, 8, 0, 63712, 77394, 20480, 4820, 612, 108, 20, 12, 4, 202904, 244013, 71244, 14968, 1852, 472, 80, 32, 4
Offset: 2

Views

Author

Scott R. Shannon, Jan 21 2023

Keywords

Comments

A circle is constructed for every pair of the n x n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the n x n points is A359931(n).
See A359932 and A359933 for images of the circles.
The first occurrence of a 2-gon is when n = 7. Assuming the grid points are separated by 1 unit, in the first quadrant this region has endpoints (6,7) and (7,6) - an equivalent region is in each of the three other quadrants. Its arcs are from two circles, one with center at (2,2) going through point (-2,-3) while the other has center (3,3) going through point (0,-1). See the attached image.

Examples

			The table begins:
0, 16, 30;
0, 412, 341, 60, 20, 4;
0, 3464, 3534, 928, 212, 48, 12;
0, 16936, 19861, 5252, 1056, 88, 52, 8;
0, 63712, 77394, 20480, 4820, 612, 108, 20, 12;
4, 202904, 244013, 71244, 14968, 1852, 472, 80, 32, 4;
.
.

Links

Crossrefs

Cf. A359932 (vertices), A359933 (regions), A359934 (edges), A359931 (distinct circles), A359862, A359258.

Formula

Sum of row n = A359933(n).

A359934 Number of edges among all distinct circles that can be constructed from an n x n square grid of points using only a compass.

Original entry on oeis.org

84, 1524, 15436, 81980, 318740, 1024312
Offset: 2

Views

Author

Scott R. Shannon, Jan 21 2023

Keywords

Comments

A circle is constructed for every pair of the n x n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the n x n points is A359931(n).
No formula for a(n) is known.
See A359932 and A359933 for images of the circles.

Crossrefs

Cf. A359932 (vertices), A359933 (regions), A359935 (k-gons), A359931 (distinct circles), A359861, A359254.

Formula

a(n) = A359932(n) + A359933(n) - 1 by Euler's formula.

A362234 Number of regions among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes when each pair of points is connected by a circle and where the points lie at the ends of the circles' diameter.

Original entry on oeis.org

32, 372, 1804, 5772, 14660, 30816, 58232, 100080, 161700, 249200, 368384
Offset: 1

Views

Author

Scott R. Shannon, Apr 13 2023

Keywords

Comments

A circle is constructed for every pair of the 1 + 4n points, the two points lying at the ends of a diameter of the circle. The number of distinct circles constructed from the points is A139275(n).
No formula for a(n) is currently known.

Links

Crossrefs

Cf. A362233 (vertices), A362235 (edges), A362236 (k-gons), A139275 (distinct circles), A353782, A359933.

Formula

a(n) = A362235(n) - A362233(n) + 1 by Euler's formula.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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