A373110 -id:A373110 - cp's OEIS frontend

A373106 Number of vertices among all 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.

5, 61, 677, 2533, 7705, 17269, 37161, 65089, 111877, 174545, 274213

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.

Scott R. Shannon, Image for n = 0. In this and other images the 4 + 4*n vertices forming the square are drawn larger for clarity.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.

Cf. A373107 (regions), A373108 (edges), A373109 (k-gons), A373110 (circles), A372977, A372731, A358746, A362233, A360351.

a(n) = A373108(n) - A373107(n) + 1 by Euler's formula.

A373108 Number of edges among all 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

16, 196, 1608, 5784, 16848, 37300, 78420, 136920, 233336, 363200, 565700

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 (vertices), A373107 (regions), A373109 (k-gons), A373110 (circles), A372979, A372733, A358783, A362235, A360353.

a(n) = A373106(n) + A373107(n) - 1 by Euler's formula.

A373107 Number of regions among all 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

12, 136, 932, 3252, 9144, 20032, 41260, 71832, 121460, 188656, 291488

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.

Scott R. Shannon, Image for n = 0. In this and other images the 4 + 4*n vertices forming the square are drawn as white dots for clarity.
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 = 7.

Cf. A373106 (vertices), A373108 (edges), A373109 (k-gons), A373110 (circles), A372978, A372732, A358782, A362234, A360352.

a(n) = A373108(n) - A373106(n) + 1 by Euler's formula.

A384700 On a 2 X n grid of vertices, draw a circle through every unordered triple of non-collinear vertices: a(n) is the number of distinct circles created.

Original entry on oeis.org

0, 1, 9, 24, 52, 93, 153, 232, 336, 465, 625, 816, 1044, 1309, 1617, 1968, 2368, 2817, 3321, 3880, 4500, 5181, 5929

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jun 07 2025

Scott R. Shannon, Image for a(3) = 9.

Cf. A384701 (vertices), A384702 (regions), A384703 (edges), A365669, A374338, A373110, A372981,

Conjecture:
for even n, a(n) = n^3/2 - n^2/4 - n,
for odd n > 1, a(n) = n^3/2 - n^2/4 - n + 3/4.

A385159 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join a circle through every unordered triple of non-collinear points: a(n) is the number of distinct circles created.

Original entry on oeis.org

1, 18, 99, 280, 633, 1098, 1915, 2928, 4329, 6010, 8331, 10752, 14113, 17778, 21987

Offset: 1

Scott R. Shannon, Jun 20 2025

Scott R. Shannon, Image for n = 2. The 4 x 2 = 8 starting points are shown as white dots.

Cf. A385160 (vertices), A385161 (regions), A385162 (edges), A361622, A384700, A373110, A372735, A365669.

A373109 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 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

8, 4, 40, 76, 20, 60, 492, 304, 56, 20, 88, 1696, 1136, 252, 64, 16, 124, 4196, 3536, 1052, 204, 28, 4, 128, 8940, 7948, 2448, 496, 68, 0, 4, 172, 16464, 17628, 5560, 1268, 164, 4, 144, 28424, 30884, 9964, 2064, 312, 24, 8, 0, 8, 196, 46844, 51840, 17832, 4112, 556, 60, 20

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.

			The table begins:
8, 4;
40, 76, 20;
60, 492, 304, 56, 20;
88, 1696, 1136, 252, 64, 16;
124, 4196, 3536, 1052, 204, 28, 4;
128, 8940, 7948, 2448, 496, 68, 0, 4;
172, 16464, 17628, 5560, 1268, 164, 4;
144, 28424, 30884, 9964, 2064, 312, 24, 8, 0, 8;
196, 46844, 51840, 17832, 4112, 556, 60, 20;
216, 71944, 80760, 28468, 6272, 856, 136, 0, 4;
264, 106588, 126856, 45148, 10780, 1628, 172, 32, 20;
.
.

Cf. A373106 (vertices), A373107 (regions), A373108 (edges), A373110 (circles), A372980, A372734, A359009, A362236, A360354.

Sum of row n = A373107(n).

cp's OEIS Frontend

A373106 Number of vertices among all 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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A373108 Number of edges among all 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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A373107 Number of regions among all 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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A384700 On a 2 X n grid of vertices, draw a circle through every unordered triple of non-collinear vertices: a(n) is the number of distinct circles created.

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Crossrefs

Formula

A385159 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join a circle through every unordered triple of non-collinear points: a(n) is the number of distinct circles created.

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Examples

Crossrefs

Formula

A373106 Number of vertices among all 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.

A373108 Number of edges among all 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.

A373107 Number of regions among all 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.