A373107 -id:A373107 - 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.

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.

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 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.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Crossrefs

Formula

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Crossrefs

Formula

Views

Author

Keywords

Comments

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.

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.