A007678 -id:A007678 - cp's OEIS frontend

A340688 Irregular table read by rows: Take a concave circular triangle with all diagonals drawn, as in A340685. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

1, 12, 22, 3, 3, 66, 36, 67, 108, 12, 222, 186, 48, 6, 265, 465, 132, 6, 582, 786, 174, 48, 732, 1905, 324, 76, 3, 6, 1410, 2268, 558, 156, 6, 1704, 3732, 861, 223, 18, 3, 2778, 4242, 1260, 324, 42, 3369, 6540, 1872, 409, 42, 24, 4896, 7302, 2502, 540, 72, 24, 6138, 10467, 3306, 907, 99, 30

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 16 2021

See A340685 for images of the regions and A340686 for images of the vertices.

			A concave circular triangle with 1 point dividing its edges, n = 2, contains 12 triangles and no other n-gons, so the second row is [12]. A concave circular triangle with 2 points dividing its edges, n = 3, contains 22 triangles, 3 quadrilaterals, 3 pentagons and no other n-gons, so the third row is [22, 3, 3].
The table begins:
1;
12;
22, 3, 3;
66, 36;
67, 108, 12;
222, 186, 48, 6;
265, 465, 132, 6;
582, 786, 174, 48;
732, 1905, 324, 76, 3, 6;
1410, 2268, 558, 156, 6;
1704, 3732, 861, 223, 18, 3;
2778, 4242, 1260, 324, 42;
3369, 6540, 1872, 409, 42, 24;
4896, 7302, 2502, 540, 72, 24;
6138, 10467, 3306, 907, 99, 30;
8364, 12522, 4566, 1020, 120, 18;
10132, 16149, 5439, 1410, 288, 57, 0, 3;
13398, 19308, 6870, 1962, 252, 30, 12;
16029, 23082, 8859, 2422, 336, 90, 3;
20682, 29658, 10800, 2976, 528, 66;

Scott R. Shannon, Image of the regions for n = 20.
Wikipedia, Circular triangle.

Cf. A340685 (regions), A340686 (vertices), A340687 (edges), A340614, A007678, A092867.

A345025 Number of regions formed when every pair of vertices of a regular n-gon are joined by an infinite line.

Original entry on oeis.org

1, 2, 7, 16, 36, 72, 141, 232, 424, 630, 1035, 1284, 2172, 2716, 4081, 4848, 7056, 7290, 11439, 12960, 17620, 19712, 26037, 26568, 37176, 40638, 51571, 55832, 69804, 64440, 92505, 98912, 120352, 128146, 154071, 156348, 194436, 205352, 242269, 254920, 298440, 290766, 363867, 380776, 439516

Offset: 1

Scott R. Shannon, Jun 06 2021

nonn

The count of regions includes both the closed bounded polygons and the open unbounded areas surrounding these polygons with two edges that go to infinity.

See A344857 for further examples and images of the regions.

			a(2) = 2 as an infinite line connecting two points cuts space into two unbounded regions.
a(3) = 7 as the three connected points of the 3-gon form one closed triangle along with six outer unbounded areas, seven regions in total.
a(4) = 16 as the four connected points of the 4-gon form four closed triangle inside the square along with twelve outer unbounded areas, sixteen regions in total.

Scott R. Shannon, Image for n = 3. In this and other images the n-gon vertices are highlighted as white dots while the outer open regions are cross-hatched. The key for the edge-number coloring is shown at the top-left of the image. Note the edge count for open areas also includes the two infinite edges.
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.

Cf. A344857 (number of polygons), A344311 (number polygons outside the n-gon), A007678 (number polygons inside the n-gon), A002378 (number of open regions for (n-1)-gon), A146212 (number of vertices), A344866, A344938.

Formula for odd n: a(n) = (n^4 - 7*n^3 + 27*n^2 - 29*n + 8)/8 (see A344857).

For n >= 3, a(n) = A344857(n) + A002378(n-1).

A217748 Number of regions with infinite area in the exterior of a regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, 208, 238, 270, 304, 340, 378, 418, 460, 504, 550, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2068, 2160, 2254, 2350, 2448, 2548

Offset: 3

Martin Renner, Mar 23 2013

nonn

For n > 3 same as A028552(n-3).

			a(3) = 1 since the equilateral triangle has no diagonals and therefore one exterior region with infinite area.
a(4) = 4 since the two diagonals of the square divide the exterior in four regions with infinite area.
a(5) = 10 since the ten diagonals of the regular pentagon divide the exterior in ten regions with infinite area of two different shapes.

Cf. A004526, A007678, A028552, A164004, A217745, A217746.

a[n_] := n*(n - 3); a[3] = 1; Array[a, 50, 3] (* Amiram Eldar, Dec 10 2022 *)

a(n) = if(n == 3, 1, n*(n-3)); \\ Amiram Eldar, Dec 10 2022

a(n) = n*(n-3) for n > 3.
a(n) = A217745(n) - A217746(n).
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=3} 1/a(n) = 29/18.
Sum_{n>=3} (-1)^(n+1)/a(n) = 23/18 - 2*log(2)/3. (End)

A260417 Number of triple-crossings of diagonals in the regular 2n-gon.

Original entry on oeis.org

0, 1, 12, 30, 128, 147, 264, 1056, 600, 825, 2380, 1482, 1932, 9635, 3024, 3672, 8484, 5301, 6300, 19474, 8580, 9867, 20744, 12900, 14664, 30141, 18564, 20706, 62200, 25575, 28320, 54956, 34272, 37485, 62868, 44622, 48564, 86359, 57000, 61500, 117068, 71337

Offset: 2

Jonathan Sondow, Jul 25 2015

nonn

Same as (total number of triangles visible in convex 2n-gon with all diagonals drawn in general position) - (total number of triangles visible in regular 2n-gon with all diagonals drawn).
Number of triple-crossings of diagonals in the regular 2n+1-gon is 0.
See Sillke 1998 (where a(n) is called "T(2n)") for explanations and extensive annotated references.
See A005732 and A006600 for more comments, references, links, formulas, examples, programs, and lists from which to compute a(n) = A005732(2n) - A006600(2n) up to n = 500.

			With only 2 diagonals in a 4-gon, there can be no triple-crossings, so a(2) = 0.

T. Sillke, Number of triangles for a convex n-gon, 1998.

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

a(n) = A005732(2n) - A006600(2n).

A332978 The number of regions formed inside a triangle with leg lengths equal to the Pythagorean triples by straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

Original entry on oeis.org

271, 5746, 14040, 32294, 50551, 108737, 180662, 276533, 259805, 558256, 591687, 901811, 1117126, 1015277, 1386667, 1223260, 1944396, 3149291, 3165147, 4523784, 4764416, 4859839, 6025266, 7186096

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 04 2020

The terms are from numeric computation - no formula for a(n) is currently known.

			The triples are ordered by the total sum of the leg lengths:
           Triple        |      Number of regions
          (3, 4, 5)      |           271
          (6, 8, 10)     |           5746
          (5, 12, 13)    |           14040
          (9, 12, 15)    |           32294
          (8, 15, 17)    |           50551
          (12, 16, 20)   |           108737
          (7, 24, 25)    |           180662
          (15, 20, 25)   |           276533
          (10, 24, 26)   |           259805
          (20, 21, 29)   |           558256
          (18, 24, 30)   |           591687
          (16, 30, 34)   |           901811
          (21, 28, 35)   |           1117126
          (12, 35, 37)   |           1015277
          (15, 36, 39)   |           1386667
          (9, 40, 41)    |           1223260
          (24, 32, 40)   |           1944396
          (27, 36, 45)   |           3149291
          (14, 48, 50)   |           3165147
          (20, 48, 52)   |           4523784
          (24, 45, 51)   |           4764416
          (30, 40, 50)   |           4859839
          (28, 45, 53)   |           6025266
          (33, 44, 55)   |           7186096

Scott R. Shannon, Triangle regions for leg lengths (3,4,5).
Scott R. Shannon, Triangle regions for leg lengths (6,8,10).
Scott R. Shannon, Triangle regions for leg lengths (5,12,13).
Scott R. Shannon, Triangle regions for leg lengths (9,12,15).
Scott R. Shannon, Triangle regions for leg lengths (8,15,17).
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Wikipedia, Pythagorean triple.

Cf. A333135 (n-gons), A333136 (vertices), A333137 (edges), A103605 (Pythagorean triple ordering), A007678, A092867, A331452.

a(8)-a(24) from Lars Blomberg, Jun 07 2020

A333135 Irregular table read by rows: Take a triangle with Pythagorean triple leg lengths with all diagonals drawn, as in A332978. Then T(n,k) = number of k-sided polygons in that figure for k >= 3 where the legs are divided into unit length parts.

Original entry on oeis.org

139, 94, 34, 3, 1, 2383, 2421, 760, 167, 13, 2, 5307, 5958, 2113, 563, 80, 17, 2, 13083, 13560, 4479, 1002, 153, 16, 1, 18827, 20896, 8256, 2139, 377, 49, 6, 1, 42992, 45400, 15930, 3771, 579, 60, 5, 63526, 79275, 28922, 7315, 1404, 202, 14, 4

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See A332978 for the Pythagorean triple ordering and the links for images of the triangles.

			Table begins:
139, 94, 34, 3, 1;
2383, 2421, 760, 167, 13, 2;
5307, 5958, 2113, 563, 80, 17, 2;
13083, 13560, 4479, 1002, 153, 16, 1;
18827, 20896, 8256, 2139, 377, 49, 6, 1;
42992, 45400, 15930, 3771, 579, 60, 5;
63526, 79275, 28922, 7315, 1404, 202, 14, 4;
The row sums are A332978.

Lars Blomberg, Table of n, a(n) for n = 1..213 (the first 24 rows)

Cf. A332978 (regions), A333136 (vertices), A333137 (edges), A103605 (Pythagorean triple ordering), A007678, A092867, A331452.

Corrected typo in a(12) and a(49) and beyond from Lars Blomberg, Jun 07 2020

A340687 The number of edges on a concave circular triangle formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

3, 21, 51, 177, 354, 852, 1686, 3036, 5994, 8550, 12933, 16938, 24303, 30192, 41616, 52581, 66600, 82857, 101199, 128409, 153669, 194505, 227193, 271635, 320898, 368703, 431640, 490884, 578805, 646869, 744717, 851214, 943068, 1071741, 1195638, 1367913, 1523694, 1713300, 1885389, 2082696

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 16 2021

nonn

The terms are from numeric computation - no formula for a(n) is currently known.

See A340685 for images of the regions and A340686 for images of the vertices.

Wikipedia, Circular triangle.

Cf. A340685 (regions), A340686 (vertices), A340688 (n-gons), A340613, A007678, A092867.

A341688 The number of regions inside a 2 by 1 ellipse formed by the straight line segments mutually connecting all points formed by dividing the ellipse into 2n equal angle sectors from its origin.

Original entry on oeis.org

0, 4, 24, 84, 232, 524, 1052, 1868, 3144, 4876, 7440, 10724, 15124, 20604, 27632, 36124, 46672, 59108, 74184, 91488, 112380, 136044, 163724, 194924, 230932, 271124, 316992, 367748, 425124, 488116, 558820, 635964, 721824, 815044, 918132, 1029524, 1152012, 1283788, 1427964, 1582328, 1750760

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 17 2021

nonn

The ellipse, with width twice its height, has points at its x-axis extrema and n-1 points both above and below the x-axis, 2n points in total. These are placed on the ellipse's perimeter by dividing it from the origin into 2n sectors of equal angle.

The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Regions for n = 3.
Scott R. Shannon, Regions for n = 5.
Scott R. Shannon, Regions for n = 9.
Scott R. Shannon, Regions for n = 19.
Scott R. Shannon, Regions for n = 24.
Scott R. Shannon, Regions for n = 13 using random distance-based coloring.
Wikipedia, Ellipse.

Cf. A341762 (vertices), A341764 (edges), A341800 (n-gons), A007678, A092867, A255011, A331929, A331931, A333075.

A341762 The number of vertices on a 2 by 1 ellipse formed by the straight line segments mutually connecting all points formed by dividing the ellipse into 2n equal angle sectors from its origin.

Original entry on oeis.org

2, 5, 19, 65, 195, 461, 971, 1737, 2995, 4617, 7203, 10385, 14779, 20125, 27155, 35481, 46051, 58277, 73395, 90323, 111403, 134765, 162539, 193385, 229515, 269301, 315331, 365617, 423195, 485617, 556603, 633145, 719299, 811845, 915275, 1025921, 1148811, 1279757, 1424395, 1577723, 1746803

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 19 2021

nonn

See A341688 for a description of the ellipse.
Curiously the only ellipses found that have vertices with three or more lines crossing that are not on the x or y axes are those with a number of vertices equal to a multiple of ten.
The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Image of the vertices for n = 3.
Scott R. Shannon, Image of the vertices for n = 5. Notice the off-axis vertices that have three lines crossing.
Scott R. Shannon, Image of the vertices for n = 9.
Scott R. Shannon, Image of the vertices for n = 10. Notice the off-axis vertices that have three or four lines crossing.
Scott R. Shannon, Image of the vertices for n = 20. Notice the off-axis vertices that have three or more lines crossing.
Scott R. Shannon, Image of the vertices for n = 21.
Wikipedia, Ellipse.

Cf. A341688 (regions), A341764 (edges), A341800 (n-gons), A007678, A092867, A255011, A331929, A331931, A333075.

A341800 Irregular table read by rows: Take a 2 by 1 ellipse with all diagonals drawn, as in A341688. Then T(n,k) = number of k-sided polygons in the figure containing 2n vertices, for k >= 3.

Original entry on oeis.org

0, 4, 18, 6, 52, 28, 4, 120, 84, 22, 6, 244, 192, 72, 16, 434, 432, 124, 54, 8, 748, 748, 300, 52, 16, 4, 1234, 1232, 482, 164, 26, 2, 4, 1896, 1940, 776, 220, 36, 8, 2764, 2926, 1332, 330, 78, 10, 3892, 4460, 1716, 536, 88, 28, 0, 4, 5580, 5918, 2642, 784, 152, 44, 4

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 20 2021

The terms are from numeric computation - no formula for a(n) is currently known.

See A341688 for a description of the ellipse and images of the regions, and A341762 for images of the vertices.

			A 2 by 1 ellipse consisting of 12 vertices, n = 6, contains 244 triangle, 192 quadrilaterals, 72 pentagons, 16 hexagons and no other n-gons, so the sixth row is [244, 192, 72, 16]. See the linked image.
The table begins:
0;
4;
18, 6;
52, 28, 4;
120, 84, 22, 6;
244, 192, 72, 16;
434, 432, 124, 54, 8;
748, 748, 300, 52, 16, 4;
1234, 1232, 482, 164, 26, 2, 4;
1896, 1940, 776, 220, 36, 8;
2764, 2926, 1332, 330, 78, 10;
3892, 4460, 1716, 536, 88, 28, 0, 4;
5580, 5918, 2642, 784, 152, 44, 4;
7508, 8204, 3540, 1108, 224, 12, 4, 4;
9902, 11202, 4636, 1472, 362, 44, 10, 4;
12984, 14508, 6208, 1920, 412, 80, 12;
16804, 18396, 8272, 2522, 522, 136, 20;
21212, 23352, 10580, 3144, 672, 112, 36;
26602, 28938, 13438, 4264, 766, 162, 12, 2;
32732, 36200, 16124, 5276, 952, 192, 12;
40026, 44216, 20038, 6564, 1302, 216, 16, 2;

Scott R. Shannon, Image of the n-gons for n = 6.
Wikipedia, Ellipse.

Cf. A341688 (regions), A341762 (vertices), A341764 (edges), A007678, A092867, A255011, A331929, A331931, A333075.

Row sums = A341688(n).

cp's OEIS Frontend

A340688 Irregular table read by rows: Take a concave circular triangle with all diagonals drawn, as in A340685. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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A345025 Number of regions formed when every pair of vertices of a regular n-gon are joined by an infinite line.

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A217748 Number of regions with infinite area in the exterior of a regular n-gon with all diagonals drawn.

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A260417 Number of triple-crossings of diagonals in the regular 2n-gon.

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A332978 The number of regions formed inside a triangle with leg lengths equal to the Pythagorean triples by straight line segments mutually connecting all vertices and all points that divide the sides into unit length parts.

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A333135 Irregular table read by rows: Take a triangle with Pythagorean triple leg lengths with all diagonals drawn, as in A332978. Then T(n,k) = number of k-sided polygons in that figure for k >= 3 where the legs are divided into unit length parts.

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A340687 The number of edges on a concave circular triangle formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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A341688 The number of regions inside a 2 by 1 ellipse formed by the straight line segments mutually connecting all points formed by dividing the ellipse into 2n equal angle sectors from its origin.

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A341762 The number of vertices on a 2 by 1 ellipse formed by the straight line segments mutually connecting all points formed by dividing the ellipse into 2n equal angle sectors from its origin.

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A341800 Irregular table read by rows: Take a 2 by 1 ellipse with all diagonals drawn, as in A341688. Then T(n,k) = number of k-sided polygons in the figure containing 2n vertices, for k >= 3.

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