A359860 -id:A359860 - cp's OEIS frontend

A353782 Number of regions 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.

112, 1264, 5548, 14976, 37092, 77096, 143560, 237504

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 currently known.

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.

Cf. A354605 (vertices), A356358 (edges), A361623 (k-gons), A361622 (distinct circles), A359933, A359860, A359253, A359570, A359046.

a(n) = A356358 - A354605(n) + 1 by Euler's formula.

A359933 Number of regions 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

45, 836, 8197, 43252, 167157, 535572

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 defines the radius. The number of distinct circles constructed from the n X n points is A359931(n).

No formula for a(n) is known.

Scott R. Shannon, Image for n = 2. In this and other images the n X n grid 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.

Cf. A359932 (vertices), A359934 (edges), A359935 (k-gons), A359931 (distinct circles), A359860, A359253.

a(n) = A359934(n) - A359932(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

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).

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

cp's OEIS Frontend

A353782 Number of regions 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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A359933 Number of regions 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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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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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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