A359009 -id:A359009 - cp's OEIS frontend

A359033 Maximum number of sides in any region when the vertices of a regular n-gon are connected by circles and where the vertices lie at the ends of the circles' diameters (cf. A359009 and A358782).

2, 3, 3, 5, 12, 7, 6, 9, 20, 11, 6, 13, 28, 15, 8, 17, 36, 19, 8, 21, 44, 23, 10, 25, 52, 27, 8, 29, 60, 31, 10, 33, 68, 35, 10, 37, 76, 39, 10, 41, 84

Offset: 2

Scott R. Shannon, Dec 12 2022

See A358782 for more images of the n-gons.

Scott R. Shannon, Image for n = 18.
Scott R. Shannon, Image for n = 28.

Cf. A359009 (k-gons), A358782 (regions).

A358782 The number of regions formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

1, 7, 12, 66, 85, 281, 264, 802, 821, 1893, 1740, 3810, 3725, 6871, 6448, 11748, 11125, 18317, 17160, 27616, 26797, 40067, 37176, 56826, 54653, 77707, 74788, 103734, 101041, 136835, 131744, 176584, 172109, 223931, 216900, 281090, 273829, 348583, 337480, 425950, 416641

Offset: 2

Scott R. Shannon, Nov 30 2022

nonn

Conjecture: for odd values of n all vertices are simple, other than those defining the diameters of the circles. No formula for n, or only the odd values of n, is currently known.
The author thanks Zach Shannon some of whose code was used in the generation of this sequence.
If n is odd, the circle containing the initial n points is not part of the graph (compare A370976-A370979). - N. J. A. Sloane, Mar 25 2024

Scott R. Shannon, Image for n = 2. In this and other images the points defining the circle diameters are show 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 = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 17.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 23.

Cf. A358746 (vertices), A358783 (edges), A359009 (k-gons), A007678, A344857.

See allso A370976-A370979.

a(n) = A358783(n) - A358746(n) + 1 by Euler's formula.

A358746 The number of vertices formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

2, 6, 5, 55, 54, 252, 169, 747, 630, 1804, 1381, 3679, 3150, 6690, 5553, 11509, 9846, 18012, 15241, 27237, 24398, 39606, 33577, 56275, 50622, 77058, 69693, 102979, 94770, 135966, 124065, 175593, 162894, 222810, 205885, 279831, 260870, 347178, 321961, 424391, 399042

Offset: 2

Scott R. Shannon, Nov 30 2022

nonn

Conjecture: for odd values of n all vertices are simple, other than those defining the diameters of the circles. No formula for n, or only the odd values of n, is currently known.
The author thanks Zach Shannon some of whose code was used in the generation of this sequence.
If n is odd, the circle containing the initial n points is not part of the graph (compare A370976-A370979). - N. J. A. Sloane, Mar 25 2024

Scott R. Shannon, Image for n = 2. In this and other images the points defining the circle diameters are show 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 = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 16.
Scott R. Shannon, Image for n = 17.

Cf. A358782 (regions), A358783 (edges), A359009 (k-gons), A007569, A146212.

See allso A370976-A370979.

a(n) = A358783(n) - A358782(n) + 1 by Euler's formula.

A358783 The number of edges formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

2, 12, 16, 120, 138, 532, 432, 1548, 1450, 3696, 3120, 7488, 6874, 13560, 12000, 23256, 20970, 36328, 32400, 54852, 51194, 79672, 70752, 113100, 105274, 154764, 144480, 206712, 195810, 272800, 255808, 352176, 335002, 446740, 422784, 560920, 534698, 695760, 659440, 850340, 815682

Offset: 2

Scott R. Shannon, Nov 30 2022

nonn

Conjecture: for odd values of n all vertices are simple, other than those defining the diameters of the circles. No formula for n, or only the odd values of n, is currently known.
See A358746 and A358782 for images of the circles.
The author thanks Zach Shannon some of whose code was used in the generation of this sequence.
If n is odd, the circle containing the initial n points is not part of the graph (compare A370976-A370979). - N. J. A. Sloane, Mar 25 2024

Cf. A358746 (vertices), A358782 (regions), A359009 (k-gons), A135565, A344899.

See allso A370976-A370979.

a(n) = A358746(n) + A358782(n) - 1 by Euler's formula.

A359061 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

Original entry on oeis.org

3, 0, 7, 0, 16, 29, 0, 30, 35, 1, 0, 90, 96, 0, 105, 126, 35, 1, 0, 272, 304, 48, 32, 0, 1, 0, 315, 324, 81, 0, 0, 0, 1, 0, 460, 940, 60, 40, 0, 0, 0, 1, 0, 671, 858, 264, 88, 11, 0, 0, 0, 1, 0, 960, 1656, 108, 48, 0, 1144, 1807, 559, 130, 13, 0, 0, 0, 0, 0, 1, 0, 1960, 3136, 448, 168, 0, 14, 0, 0, 0, 0, 0, 1

Offset: 2

Scott R. Shannon, Dec 14 2022

See A331702 and A359046 for further details and images.

Conjecture: the only value for n which leads to the creation of 2-gons is n = 2. Despite values for n mod 6 = 0 forming intersecting arcs at the center of the n-gon, these are cut by other circles and thus create 3-gons or 4-gons. This is in contrast to values of n mod 4 = 0 in A359009 which do lead to the creation of 2-gons at the center of the figure from similar arcs.

			The table begins:
3;
0, 7;
0, 16, 29;
0, 30, 35, 1;
0, 90, 96;
0, 105, 126, 35, 1;
0, 272, 304, 48, 32, 0, 1;
0, 315, 324, 81, 0, 0, 0, 1;
0, 460, 940, 60, 40, 0, 0, 0, 1;
0, 671, 858, 264, 88, 11, 0, 0, 0, 1;
0, 960, 1656, 108, 48;
0, 1144, 1807, 559, 130, 13, 0, 0, 0, 0, 0, 1;
0, 1960, 3136, 448, 168, 0, 14, 0, 0, 0, 0, 0, 1;
0, 2100, 3270, 945, 180, 15, 0, 0, 0, 0, 0, 0, 0, 1;
0, 3088, 5584, 896, 368, 16, 16, 0, 0, 0, 0, 0, 0, 0, 1;
0, 3400, 5814, 1513, 493, 85, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 4536, 8712, 1224, 288, 54, 36;
0, 5586, 8797, 2774, 665, 76, 152, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 7940, 12480, 2440, 960, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 7833, 14175, 3486, 1050, 147, 63, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 10428, 19448, 3850, 1408, 22, 44, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

Scott R. Shannon, Image for n = 17.
Scott R. Shannon, Image for n = 23.
Scott R. Shannon, Image for n = 24.

Cf. A331702 (vertices), A359046 (regions), A359047 (edges), A359009, A358782, A007678.

Sum of row n = A359046(n).

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);

A371376 Irregular table read by rows: place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. T(n,k), k>=2, gives the number of k-sided regions formed.

Original entry on oeis.org

1, 6, 3, 8, 0, 1, 15, 30, 5, 1, 18, 30, 14, 147, 35, 7, 7, 1, 8, 152, 48, 8, 0, 0, 1, 27, 351, 171, 36, 27, 10, 390, 200, 10, 40, 0, 0, 0, 1, 22, 693, 649, 33, 77, 0, 0, 0, 0, 1, 12, 780, 408, 0, 48, 26, 1404, 1183, 234, 169, 0, 0, 0, 0, 0, 0, 1, 14, 1498, 1274, 224, 154, 14, 14, 0, 0, 0, 0, 0, 1

Offset: 2

Scott R. Shannon, Mar 20 2024

See A371373 and A371374 for images of the graphs.

			The table begins:
1;
6, 3;
8, 0, 1;
15, 30, 5, 1;
18, 30;
14, 147, 35, 7, 7, 1;
8, 152, 48, 8, 0, 0, 1;
27, 351, 171, 36, 27;
10, 390, 200, 10, 40, 0, 0, 0, 1;
22, 693, 649, 33, 77, 0, 0, 0, 0, 1;
12, 780, 408, 0, 48;
26, 1404, 1183, 234, 169, 0, 0, 0, 0, 0, 0, 1;
14, 1498, 1274, 224, 154, 14, 14, 0, 0, 0, 0, 0, 1;
45, 2310, 2400, 390, 255, 15;
16, 2736, 2032, 656, 320, 0, 32, 0, 0, 0, 0, 0, 0, 0, 1;
34, 3978, 4097, 969, 493, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1;
18, 4410, 3078, 972, 468, 36, 18;
76, 6365, 6365, 1596, 855, 95, 76, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
20, 6840, 6000, 2100, 780, 60, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
63, 8946, 10395, 2751, 924, 126, 147;
22, 10076, 9218, 3674, 1166, 22, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
46, 13156, 14996, 4347, 1702, 92, 138, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                                                                     \\ 0, 0, 1;
24, 14232, 13296, 4512, 1440, 96, 240;
100, 19075, 19850, 6975, 2675, 150, 175, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                                                                  \\ 0, 0, 0, 1;
.
.

Cf. A371373 (vertices), A371374 (regions), A371375 (edges), A371377 (vertex crossings), A371274, A331450, A359009, A359061.

Sum of row(n) = A371374(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).

A373109 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 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

8, 4, 40, 76, 20, 60, 492, 304, 56, 20, 88, 1696, 1136, 252, 64, 16, 124, 4196, 3536, 1052, 204, 28, 4, 128, 8940, 7948, 2448, 496, 68, 0, 4, 172, 16464, 17628, 5560, 1268, 164, 4, 144, 28424, 30884, 9964, 2064, 312, 24, 8, 0, 8, 196, 46844, 51840, 17832, 4112, 556, 60, 20

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.

			The table begins:
8, 4;
40, 76, 20;
60, 492, 304, 56, 20;
88, 1696, 1136, 252, 64, 16;
124, 4196, 3536, 1052, 204, 28, 4;
128, 8940, 7948, 2448, 496, 68, 0, 4;
172, 16464, 17628, 5560, 1268, 164, 4;
144, 28424, 30884, 9964, 2064, 312, 24, 8, 0, 8;
196, 46844, 51840, 17832, 4112, 556, 60, 20;
216, 71944, 80760, 28468, 6272, 856, 136, 0, 4;
264, 106588, 126856, 45148, 10780, 1628, 172, 32, 20;
.
.

Cf. A373106 (vertices), A373107 (regions), A373108 (edges), A373110 (circles), A372980, A372734, A359009, A362236, A360354.

Sum of row n = A373107(n).

cp's OEIS Frontend

A359033 Maximum number of sides in any region when the vertices of a regular n-gon are connected by circles and where the vertices lie at the ends of the circles' diameters (cf. A359009 and A358782).

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A358782 The number of regions formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

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A358746 The number of vertices formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

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A358783 The number of edges formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

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A359061 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, among all circles that can be constructed on vertices of an n-sided regular polygon, 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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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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