A359931 -id:A359931 - cp's OEIS frontend

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.

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).

A360350 Number of 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, 26, 79, 185, 366, 653, 1077, 1678, 2494, 3571, 4959, 6718, 8889, 11541, 14740, 18553, 23027, 28278, 34351, 41352, 49356, 58454, 68732, 80330, 93304, 107757, 123815, 141605, 161211, 182795, 206393, 232190, 260331, 290907, 324090, 360080, 398856, 440655, 485655

Offset: 2

Scott R. Shannon, Feb 03 2023

nonn

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.
No formula for a(n) is known.
See A360351 and A360352 for images of the resulting vertices and regions.

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. A360351 (vertices), A360352 (regions), A360353 (edges), A360354 (k-gons), A359931.

a(n) = { my (p = vector(n^2, k, (k-1)%n + ((k-1)\n)*I)); #setbinop((i,j)->[i+j, norm(i-j)], p)-n^2; } \\ Rémy Sigrist, Sep 24 2023

More terms from Rémy Sigrist, Sep 24 2023

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.

A365669 Number of distinct circles created after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex.

Original entry on oeis.org

0, 1, 2, 6, 114, 42103152

Offset: 1

Scott R. Shannon, Sep 15 2023

See A359569 for further details and images.

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. A359569 (vertices), A359570 (regions), A359571 (edges), A359619 (k-gons), A359931, A360350, A361622.

A372682 Number of distinct circles that can be constructed from the 3 vertices and the equally spaced 3*n points placed on the sides of an equilateral triangle, using only a compass.

Original entry on oeis.org

3, 15, 36, 69, 123, 180, 264, 339, 453, 549, 702, 807, 999, 1128, 1329, 1494, 1749, 1935, 2214, 2373, 2682, 2940, 3288, 3483

Offset: 0

Scott R. Shannon, May 10 2024

See A372614 for images of the circles.

Cf. A372614, A372615, A359931, A360350, A361622, A365669.

A372735 Number of distinct circles that can be constructed from the 3 vertices and the equally spaced 3n points placed on the sides of an equilateral triangle when every pair of the 3 + 3n points are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

3, 15, 34, 63, 99, 148, 201, 267, 340, 423, 513, 616, 723, 843, 970, 1107, 1251, 1408, 1569, 1743, 1924, 2115, 2313, 2524, 2739, 2967, 3202, 3447, 3699

Offset: 1

Scott R. Shannon, May 11 2024

See A372731 for images of the circles.

Cf. A372731, A372732, A372682, A372614, A372615, A359931, A360350, A361622, A365669.

A372981 Number of distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square, using only a compass.

Original entry on oeis.org

8, 32, 88, 160, 264, 400, 576, 732, 968, 1184, 1480, 1728, 2104, 2424, 2840, 3196, 3688, 4088, 4640, 5048, 5704, 6248, 6904, 7364

Offset: 0

Scott R. Shannon, May 19 2024

A circle is constructed for every pair of the 4 + 4*n points, the first point defines the circle's center while the second the radius distance.

See A372978 for images of the circles.

Cf. A372978, A372977, A372979, A359931, A372682, A361622, A365669.

A373110 Number of distinct circles that can be constructed from the 4 vertices and the equally spaced 4n points placed on the sides of a square when every pair of the 4 + 4n points are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

5, 22, 54, 99, 159, 232, 320, 421, 537, 666, 810, 967, 1139, 1324, 1524, 1737, 1965, 2206, 2462, 2731, 3015, 3312, 3624, 3949

Offset: 0

Scott R. Shannon, May 25 2024

A circle is constructed for every pair of the 4 + 4*n points, the two points lying at the ends of a diameter of the circle.

See A373106 and A373107 for images of the circles.

Cf. A373106, A373107, A373108, A372981, A372735, A359931, A360350, A361622, A365669.

Conjectured:
For even n, a(n) = (14*n^2 + 21*n + 10)/2.
For odd n, a(n) = (14*n^2 + 21*n + 9)/2.

cp's OEIS Frontend

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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A360350 Number of 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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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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A365669 Number of distinct circles created after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex.

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A372682 Number of distinct circles that can be constructed from the 3 vertices and the equally spaced 3*n points placed on the sides of an equilateral triangle, using only a compass.

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A372735 Number of distinct circles that can be constructed from the 3 vertices and the equally spaced 3*n points placed on the sides of an equilateral triangle when every pair of the 3 + 3*n points are connected by a circle and where the points lie at the ends of the circle's diameter.

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A372981 Number of distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square, using only a compass.

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A373110 Number of distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square when every pair of the 4 + 4*n points are connected by a circle and where the points lie at the ends of the circle's diameter.

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A372735 Number of distinct circles that can be constructed from the 3 vertices and the equally spaced 3n points placed on the sides of an equilateral triangle when every pair of the 3 + 3n points are connected by a circle and where the points lie at the ends of the circle's diameter.

A373110 Number of distinct circles that can be constructed from the 4 vertices and the equally spaced 4n points placed on the sides of a square when every pair of the 4 + 4n points are connected by a circle and where the points lie at the ends of the circle's diameter.