A359934 -id:A359934 - 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).

A356358 Number of edges 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

212, 2408, 10548, 28728, 71588, 149280, 278716, 461824

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 known.
See A354605 and A353782 for images of the vertices and regions.

Cf. A354605 (vertices), A353782 (regions), A361623 (k-gons), A361622 (distinct circles), A359934, A359861, A359254, A359571, A359047.

a(n) = A353782(n) + A354605(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

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

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

a(n) = A359934(n) - A359933(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).

A360353 Number of edges 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

16, 244, 2580, 15788, 67144, 227888, 668008, 1712960, 3968028

Offset: 2

Scott R. Shannon, Feb 04 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).

See A360351 and A360352 for images of the circles.

Cf. A360351 (vertices), A360352 (regions), A360354 (k-gons), A360350 (distinct circles), A359934.

a(n) = A360351(n) + A360352(n) - 1 by Euler's formula.

A362235 Number of edges 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

48, 620, 3184, 10516, 27240, 57676, 109880, 189436, 307200, 474820, 703880

Offset: 1

Scott R. Shannon, Apr 13 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.
See A362233 and A362234 for images of the circles.

Cf. A362233 (vertices), A362234 (regions), A362236 (k-gons), A139275 (distinct circles), A356358, A359934.

a(n) = A362234(n) + A362233(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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A356358 Number of edges 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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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.

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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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A360353 Number of edges 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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A362235 Number of edges 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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