A359861 -id:A359861 - cp's OEIS frontend

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.

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.

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.

A359859 Number of vertices 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

2, 40, 190, 740, 1824, 3956, 7314, 12956, 20684, 32276, 47348, 68516, 94550, 128780, 170106, 222252, 283418, 358756, 445534, 550868, 670358, 811556, 970740, 1157168, 1363700, 1601384, 1864524, 2164668, 2493136, 2865176, 3269606, 3724112, 4215536, 4762284, 5353050

Offset: 1

Scott R. Shannon, Jan 16 2023

nonn

A circle is constructed for every pair of the 2 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 2 x n points is conjectured to be 4*A001859(n-1).

No formula for a(n) is known.

Lucas A. Brown, Python program.
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 = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.

Cf. A359860 (regions), A359861 (edges), A359862 (k-gons), A001859, A359252.

a(n) = A359861(n) - A359860(n) + 1 by Euler's formula.

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

A359860 Number of regions 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

3, 45, 231, 865, 2081, 4489, 8211, 14401, 22857, 35445, 51741, 74397, 102271, 138801, 182739, 238181, 303175, 383097, 474995, 586021, 712003, 860829, 1028225, 1223773, 1440593, 1689993, 1965525, 2279509, 2622993, 3011405, 3433615, 3907241, 4419261, 4988781, 5603271

Offset: 1

Scott R. Shannon, Jan 16 2023

nonn

See A359859 for further details. No formula for a(n) is known.

Lucas A. Brown, Python program.
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 = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 8.

Cf. A359859 (vertices), A359861 (edges), A359862 (k-gons), A001859, A359253.

a(n) = A359861(n) - A359859(n) + 1 by Euler's formula.

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

A359862 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 a 2 x n square grid of points using only a compass.

Original entry on oeis.org

3, 0, 16, 29, 0, 102, 117, 10, 2, 4, 368, 402, 64, 26, 1, 12, 860, 903, 252, 52, 0, 2, 12, 1812, 2028, 520, 110, 4, 3, 24, 3168, 3841, 960, 204, 8, 6, 32, 5420, 6804, 1748, 362, 24, 11, 44, 8388, 10987, 2826, 552, 46, 14, 56, 12808, 17122, 4448, 922, 72, 17, 64, 18348, 25257, 6594, 1370, 82, 26

Offset: 1

Scott R. Shannon, Jan 16 2023

See A359859 and A359860 for further details and images of the circles.

			The table begins:
   3;
   0,    16,    29;
   0,   102,   117,   10,    2;
   4,   368,   402,   64,   26,  1;
  12,   860,   903,  252,   52,  0,  2;
  12,  1812,  2028,  520,  110,  4,  3;
  24,  3168,  3841,  960,  204,  8,  6;
  32,  5420,  6804, 1748,  362, 24, 11;
  44,  8388, 10987, 2826,  552, 46, 14;
  56, 12808, 17122, 4448,  922, 72, 17;
  64, 18348, 25257, 6594, 1370, 82, 26;
  ...

Scott R. Shannon, Image for n = 7.

Cf. A359859 (vertices), A359860 (regions), A359861 (edges), A001859, A359258.

Sum of row n = A359860(n).

cp's OEIS Frontend

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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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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A359859 Number of vertices 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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A359860 Number of regions 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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A359862 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 a 2 x n square grid of points using only a compass.

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