A359932 -id:A359932 - cp's OEIS frontend

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

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

Offset: 2

Scott R. Shannon, Jan 19 2023

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.

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

A354605 Number of vertices 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

101, 1145, 5001, 13753, 34497, 72185, 135157, 224321

Offset: 1

Scott R. Shannon, Mar 13 2023

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.

Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.

Cf. A353782 (regions), A356358 (edges), A361623 (k-gons), A361622 (distinct circles), A359932, A359859, A359252, A359569, A331702.

a(n) = A356358 - A353782(n) + 1 by Euler's formula.

A360351 Number of vertices 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

5, 77, 1045, 6885, 30265, 104421, 309973, 800185, 1862053

Offset: 2

Scott R. Shannon, Feb 03 2023

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).

Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
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.)

Cf. A360352 (regions), A360353 (edges), A360354 (k-gons), A360350 (distinct circles), A359932.

a(n) = A360353(n) - A360352(n) + 1 by Euler's formula.

A359933 Number of regions 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

45, 836, 8197, 43252, 167157, 535572

Offset: 2

Scott R. Shannon, Jan 21 2023

A circle is constructed for every pair of the n X n points; the first point defines the circle's center while the second defines the radius. The number of distinct circles constructed from the n X n points is A359931(n).

No formula for a(n) is known.

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.

Cf. A359932 (vertices), A359934 (edges), A359935 (k-gons), A359931 (distinct circles), A359860, A359253.

a(n) = A359934(n) - A359932(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

Scott R. Shannon, Jan 21 2023

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.

			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;
.
.

Scott R. Shannon, Close-up of the circles for n = 7. The 2-gon is the thin region shown in white in the upper-right corner.

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

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

Scott R. Shannon, Jan 21 2023

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.

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

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

A362233 Number of vertices 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

17, 249, 1381, 4745, 12581, 26861, 51649, 89357, 145501, 225621, 335497

Offset: 1

Scott R. Shannon, Apr 12 2023

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.

Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.

Cf. A362234 (regions), A362235 (edges), A362236 (k-gons), A139275 (distinct circles), A354605, A359932.

a(n) = A362235(n) - A362234(n) + 1 by Euler's formula.

cp's OEIS Frontend

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

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A354605 Number of vertices 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.

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A360351 Number of vertices 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.

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A359933 Number of regions among all distinct circles that can be constructed from an n X n square grid of points using only a compass.

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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.

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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.

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A362233 Number of vertices 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.

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