A359258 -id:A359258 - 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.

A359254 Number of edges 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

4, 26, 96, 216, 480, 882, 1564, 2464, 3804, 5502, 7912, 10864, 14816, 19526, 25508, 32392, 40992, 50818, 62852, 76432, 92460, 110590, 131904, 155306, 182316, 212188, 246316, 283586, 326100

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).

See A359252 and A359253 for images of the circles.

Cf. A359252 (vertices), A359253 (regions), A359258 (k-gons), A001859, A290866, A359047, A358783.

a(n) = A359252(n) + A359253(n) - 1 by Euler's formula.

A359935 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 an n x n square grid of points using only a compass.

Original entry on oeis.org

0, 16, 30, 0, 412, 341, 60, 20, 4, 0, 3464, 3534, 928, 212, 48, 12, 0, 16936, 19861, 5252, 1056, 88, 52, 8, 0, 63712, 77394, 20480, 4820, 612, 108, 20, 12, 4, 202904, 244013, 71244, 14968, 1852, 472, 80, 32, 4

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).
See A359932 and A359933 for images of the circles.
The first occurrence of a 2-gon is when n = 7. Assuming the grid points are separated by 1 unit, in the first quadrant this region has endpoints (6,7) and (7,6) - an equivalent region is in each of the three other quadrants. Its arcs are from two circles, one with center at (2,2) going through point (-2,-3) while the other has center (3,3) going through point (0,-1). See the attached image.

			The table begins:
0, 16, 30;
0, 412, 341, 60, 20, 4;
0, 3464, 3534, 928, 212, 48, 12;
0, 16936, 19861, 5252, 1056, 88, 52, 8;
0, 63712, 77394, 20480, 4820, 612, 108, 20, 12;
4, 202904, 244013, 71244, 14968, 1852, 472, 80, 32, 4;
.
.

Scott R. Shannon, Close-up of the circles for n = 7. The 2-gon is the thin region shown in white in the upper-right corner.

Cf. A359932 (vertices), A359933 (regions), A359934 (edges), A359931 (distinct circles), A359862, A359258.

Sum of row n = A359933(n).

A361623 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 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

0, 40, 60, 12, 0, 484, 583, 160, 28, 8, 0, 2196, 2416, 804, 104, 28, 0, 5676, 6616, 2184, 460, 40, 8, 13456, 16936, 5236, 1340, 104, 12, 4, 27512, 35032, 11796, 2400, 320, 28, 0, 4, 0, 50688, 65044, 22536, 4632, 584, 60, 12, 4, 8, 84300, 105860, 38024, 8124, 1080, 108

Offset: 1

Scott R. Shannon, Mar 18 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).

See A354605 and A353782 for images of the vertices and regions.

			The table begins:
 0, 40, 60, 12;
 0, 484, 583, 160, 28, 8;
 0, 2196, 2416, 804, 104, 28;
 0, 5676, 6616, 2184, 460, 40;
 8, 13456, 16936, 5236, 1340, 104, 12;
 4, 27512, 35032, 11796, 2400, 320, 28, 0, 4;
 0, 50688, 65044, 22536, 4632, 584, 60, 12, 4;
 8, 84300, 105860, 38024, 8124, 1080, 108;
.
.

Cf. A354605 (vertices), A353782 (regions), A356358 (edges), A361622 (distinct circles), A359935, A359862, A359258, A359619, A359061.

Sum of row n = A353782(n).

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).

A359619 Irregular table read by rows: T(n,k) is the number of k-gons, k>=1, 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, 0, 0, 2, 1, 0, 1, 16, 4, 0, 16, 2470, 3599, 902, 168, 14

Offset: 1

Scott R. Shannon, Jan 07 2023

See A359569 and A359570 for further details and images.

			The table begins:
0;
1;
0, 0, 2, 1;
0, 1, 16, 4;
0, 16, 2470, 3599, 902, 168, 14;
.
.

Cf. A359569 (vertices), A359570 (regions), A359571 (edges), A359258, A359061, A359009.

Sum of row n = A359570(n);

A371274 Irregular table read by rows: T(n,k) is the number of k-sided regions, k>=2, 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, 3, 3, 4, 0, 1, 10, 10, 5, 1, 12, 6, 14, 56, 21, 0, 7, 1, 8, 48, 32, 0, 0, 0, 1, 27, 144, 54, 27, 18, 10, 160, 70, 0, 30, 0, 0, 0, 1, 22, 253, 330, 11, 33, 0, 0, 0, 0, 1, 12, 276, 204, 0, 24, 26, 624, 403, 130, 104, 0, 0, 0, 0, 0, 0, 1, 14, 630, 448, 112, 70, 14, 14, 0, 0, 0, 0, 0, 1, 45, 960, 915, 165, 165

Offset: 2

Scott R. Shannon, Mar 17 2024

See A371253 and A371254 for images.

			The table begins:
1;
3, 3;
4, 0, 1;
10, 10, 5, 1;
12, 6;
14, 56, 21, 0, 7, 1;
8, 48, 32, 0, 0, 0, 1;
27, 144, 54, 27, 18;
10, 160, 70, 0, 30, 0, 0, 0, 1;
22, 253, 330, 11, 33, 0, 0, 0, 0, 1;
12, 276, 204, 0, 24;
26, 624, 403, 130, 104, 0, 0, 0, 0, 0, 0, 1;
14, 630, 448, 112, 70, 14, 14, 0, 0, 0, 0, 0, 1;
45, 960, 915, 165, 165;
16, 1136, 704, 272, 192, 0, 16, 0, 0, 0, 0, 0, 0, 0, 1;
34, 1581, 1870, 238, 272, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1;
18, 1656, 1386, 270, 288, 0, 18;
38, 2622, 2546, 646, 513, 38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
20, 2680, 2420, 820, 380, 20, 60, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
63, 3297, 4725, 1050, 315, 42, 105;
22, 3696, 4136, 1342, 484, 22, 66, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

Cf. A371253 (regions), A371254 (vertices), A371255 (edges), A331450, A359009, A359061, A359258.

Sum of row n = A371253(n).

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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A359254 Number of edges among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

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A359935 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 an n x n square grid of points using only a compass.

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A361623 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 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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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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A359619 Irregular table read by rows: T(n,k) is the number of k-gons, k>=1, 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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A371274 Irregular table read by rows: T(n,k) is the number of k-sided regions, k>=2, 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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