A372981 -id:A372981 - cp's OEIS frontend

A372977 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, using only a compass.

40, 553, 4204, 14505, 39004, 94365, 197464, 320925, 569600

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.

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. A372978 (regions), A372979 (edges), A372980 (k-gons), A372981 (circles), A372614, A372731, A371373, A354605, A360351.

a(n) = A372979(n) - A372978(n) + 1 by Euler's formula.

A372978 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, using only a compass.

Original entry on oeis.org

45, 628, 4633, 15476, 41561, 98808, 206317, 333272, 590181

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.

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.

Cf. A372977 (vertices), A372979 (edges), A372980 (k-gons), A372981 (circles), A372615, A371374, A353782, A360352.

a(n) = A372979(n) - A372977(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.

A372979 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, using only a compass.

Original entry on oeis.org

84, 1180, 8836, 29980, 80564, 193172, 403780, 654196, 1159780

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 A372977 and A372978 for images of the circles.

Cf. A372977 (vertices), A372978 (regions), A372980 (k-gons), A372981 (circles), A372616, A371375, A356358, A360353.

a(n) = A372977(n) + A372978(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.

A372980 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, 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, using only a compass.

Original entry on oeis.org

16, 29, 272, 260, 80, 12, 4, 1708, 2253, 528, 120, 20, 4, 5200, 7636, 2136, 432, 44, 20, 8, 13732, 20788, 5712, 1120, 184, 17, 8, 31576, 49284, 14060, 3180, 584, 108, 16, 64748, 103557, 30372, 6472, 980, 172, 4, 12, 103368, 166804, 49920, 11196, 1660, 260, 48, 16, 181376, 296388, 88916, 19844, 3128, 445, 64, 20

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.
Unlike A372617, a similar sequence but with vertices on an equilateral triangle, for the terms studied no graph has 2-edged regions.
See A372977 and A372978 for images of the circles.

			The table begins:
16, 29;
272, 260, 80, 12, 4;
1708, 2253, 528, 120, 20, 4;
5200, 7636, 2136, 432, 44, 20, 8;
13732, 20788, 5712, 1120, 184, 17, 8;
31576, 49284, 14060, 3180, 584, 108, 16;
64748, 103557, 30372, 6472, 980, 172, 4, 12;
103368, 166804, 49920, 11196, 1660, 260, 48, 16;
181376, 296388, 88916, 19844, 3128, 445, 64, 20;
.
.

Cf. A372977 (vertices), A372978 (regions), A372979 (edges), A372981 (circles), A372617, A371376, A361623, A360354.

Sum of row n = A372978(n).

cp's OEIS Frontend

A372977 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, using only a compass.

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A372978 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, 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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A372979 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, using only a compass.

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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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A372980 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, 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, 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 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.