A359254 -id:A359254 - cp's OEIS frontend

A359252 Number of vertices among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

2, 13, 46, 101, 226, 417, 744, 1169, 1802, 2599, 3742, 5139, 7022, 9261, 12110, 15367, 19456, 24117, 29858, 36323, 43950, 52595, 62784, 73931, 86806, 101059, 117364, 135155, 155506

Offset: 2

Scott R. Shannon, Dec 22 2022

A circle is constructed for every pair of the n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed for n points is A001859(n-1).

No formula for a(n) is currently known.

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.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 20.

Cf. A359253 (regions), A359254 (edges), A359258 (k-gons), A001859, A290447, A331702, A358746.

a(n) = A359254(n) - A359253(n) + 1 by Euler's formula.

A359253 Number of regions among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

Original entry on oeis.org

3, 14, 51, 116, 255, 466, 821, 1296, 2003, 2904, 4171, 5726, 7795, 10266, 13399, 17026, 21537, 26702, 32995, 40110, 48511, 57996, 69121, 81376, 95511, 111130, 128953, 148432, 170595

Offset: 2

Scott R. Shannon, Dec 22 2022

A circle is constructed for every pair of the n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed for n points is A001859(n-1).

No formula for a(n) is currently known.

Scott R. Shannon, Image for n = 2. In this and other images the n 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.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 20.

Cf. A359252 (vertices), A359254 (edges), A359258 (k-gons), A001859, A290865, A359046, A358782.

a(n) = A359254(n) - A359252(n) + 1 by Euler's formula.

A359258 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 n equally spaced points along a line using only a compass.

Original entry on oeis.org

3, 0, 8, 4, 2, 0, 22, 23, 4, 2, 0, 50, 52, 12, 2, 0, 110, 103, 36, 6, 0, 190, 200, 64, 12, 0, 314, 387, 88, 28, 4, 0, 498, 606, 152, 32, 8, 0, 770, 941, 228, 58, 4, 2, 0, 1132, 1352, 338, 68, 12, 2, 0, 1602, 1935, 532, 98, 4, 0, 2122, 2798, 684, 106, 16, 0, 2850, 3843, 940, 132, 24, 6

Offset: 2

Scott R. Shannon, Dec 23 2022

A circle is constructed for every pair of the n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed for n points is A001859(n-1).

See A359252 and A359253 for other images of the circles.

			The table begins:
  3;
  0,     8,     4,    2;
  0,    22,    23,    4,   2;
  0,    50,    52,   12,   2;
  0,   110,   103,   36,   6;
  0,   190,   200,   64,  12;
  0,   314,   387,   88,  28,  4;
  0,   498,   606,  152,  32,  8;
  0,   770,   941,  228,  58,  4,  2;
  0,  1132,  1352,  338,  68, 12,  2;
  0,  1602,  1935,  532,  98,  4;
  0,  2122,  2798,  684, 106, 16;
  0,  2850,  3843,  940, 132, 24,  6;
  0,  3774,  4998, 1268, 192, 28,  6;
  0,  4950,  6475, 1644, 276, 44, 10;
  0,  6190,  8454, 1978, 326, 74,  4;
  0,  7778, 10737, 2520, 434, 52, 12, 4;
  0,  9674, 13224, 3202, 528, 58, 12, 4;
  0, 11978, 16169, 4116, 640, 68, 20, 4;
  ...

Scott R. Shannon, Image for n = 12.

Cf. A359253 (regions), A359252 (vertices), A359254 (edges), A001859, A332723, A359061, A359009.

Sum of row n = A359253(n);

A356358 Number of edges among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes using only a compass.

Original entry on oeis.org

212, 2408, 10548, 28728, 71588, 149280, 278716, 461824

Offset: 1

Scott R. Shannon, Mar 13 2023

A circle is constructed for every pair of the 1 + 4n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the points is A361622(n).
No formula for a(n) is known.
See A354605 and A353782 for images of the vertices and regions.

Cf. A354605 (vertices), A353782 (regions), A361623 (k-gons), A361622 (distinct circles), A359934, A359861, A359254, A359571, A359047.

a(n) = A353782(n) + A354605(n) - 1 by Euler's formula.

A359571 Number of (curved) edges after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex. See the Comments.

Original entry on oeis.org

0, 1, 6, 34, 13730

Offset: 1

Scott R. Shannon, Jan 06 2023

See A359569 and A359570 for further details and images.

Cf. A359569 (vertices), A359570 (regions), A359619 (k-gons), A359254, A359047, A358783.

For n >= 3, a(n) = A359569(n) + A359570(n) - 1 by Euler's formula.

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.

A359861 Number of edges among all distinct circles that can be constructed from a 2 X n square grid of points using only a compass.

Original entry on oeis.org

4, 84, 420, 1604, 3904, 8444, 15524, 27356, 43540, 67720, 99088, 142912, 196820, 267580, 352844, 460432, 586592, 741852, 920528, 1136888, 1382360, 1672384, 1998964, 2380940, 2804292, 3291376, 3830048, 4444176, 5116128, 5876580, 6703220, 7631352, 8634796, 9751064, 10956320

Offset: 1

Scott R. Shannon, Jan 16 2023

nonn

See A359859 and A359860 for further details and images of the circles. No formula for a(n) is known.

			a(n) = A359859(n) + A359860(n) - 1 by Euler's formula.

Lucas A. Brown, Python program.

Cf. A359859 (vertices), A359860 (regions), A359862 (k-gons), A001859, A359254.

a(19)-a(35) from Lucas A. Brown, Oct 11 2024

A371255 Number of (curved) edges formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

Original entry on oeis.org

1, 2, 9, 8, 40, 24, 168, 152, 477, 490, 1199, 912, 2418, 2464, 4230, 4464, 7769, 6894, 12369, 12400, 18606, 19008, 27784, 26376, 39575, 39390, 54027, 54432, 73254, 68340, 96410, 96800, 124443, 125222, 159005, 156168, 200540, 201932, 248508, 250120

Offset: 1

Scott R. Shannon, Mar 16 2024

nonn

See A371253 and A371254 for images of the circles.

Cf. A371253 (regions), A371254 (vertices), A371274 (k-gons), A135565, A358783, A359047, A359254.

a(n) = A371253(n) + A371254(n) - 1 by Euler's formula.

cp's OEIS Frontend

A359252 Number of vertices among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

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A359253 Number of regions among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

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A359258 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 n equally spaced points along a line using only a compass.

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A356358 Number of edges among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes using only a compass.

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A359571 Number of (curved) edges after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex. See the Comments.

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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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A359861 Number of edges among all distinct circles that can be constructed from a 2 X n square grid of points using only a compass.

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A371255 Number of (curved) edges formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

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