A360350 -id:A360350 - cp's OEIS frontend

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.

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.

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

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).

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.)

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

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

A360354 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 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

8, 4, 40, 108, 20, 92, 904, 456, 76, 0, 8, 200, 4540, 3400, 652, 100, 4, 8, 404, 17244, 15324, 3148, 628, 116, 16, 528, 54252, 54476, 11672, 2152, 332, 44, 12, 972, 151992, 158468, 37244, 7940, 1120, 224, 48, 12, 16, 1404, 379488, 404148, 103436, 20216, 3316, 600, 132, 20, 16

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.

			The table begins:
8, 4;
40, 108, 20;
92, 904, 456, 76, 0, 8;
200, 4540, 3400, 652, 100, 4, 8;
404, 17244, 15324, 3148, 628, 116, 16;
528, 54252, 54476, 11672, 2152, 332, 44, 12;
972, 151992, 158468, 37244, 7940, 1120, 224, 48, 12, 16;
1404, 379488, 404148, 103436, 20216, 3316, 600, 132, 20, 16;
1896, 868460, 923656, 252096, 49848, 7916, 1744, 276, 84;
.
.

Cf. A360351 (vertices), A360352 (regions), A360353 (edges), A360354 (k-gons), A360350 (distinct circles), A359935.

Sum of row n = A360352(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.

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.

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

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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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.

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A360354 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 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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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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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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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.