A092867 -id:A092867 - cp's OEIS frontend

A340685 The number of regions inside 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.

1, 12, 28, 102, 187, 462, 868, 1590, 3046, 4398, 6541, 8646, 12256, 15336, 20947, 26610, 33478, 41832, 50821, 64710, 77110, 97878, 113932, 136560, 160849, 185220, 216286, 246450, 289945, 324594, 372976, 426936, 472231, 537366, 598606, 685650, 762736, 858546, 943684, 1043442, 1143751, 1258800

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.

Scott R. Shannon, Regions for n = 2.
Scott R. Shannon, Regions for n = 3.
Scott R. Shannon, Regions for n = 4.
Scott R. Shannon, Regions for n = 6.
Scott R. Shannon, Regions for n = 8.
Scott R. Shannon, Regions for n = 10.
Scott R. Shannon, Regions for n = 14.
Scott R. Shannon, Regions for n = 15.
Wikipedia, Circular triangle.

Cf. A340686 (vertices), A340687 (edges), A340688 (n-gons), A340639, A007678, A092867.

A340686 The number of vertices 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, 10, 24, 76, 168, 391, 819, 1447, 2949, 4153, 6393, 8293, 12048, 14857, 20670, 25972, 33123, 41026, 50379, 63700, 76560, 96628, 113262, 135076, 160050, 183484, 215355, 244435, 288861, 322276, 371742, 424279, 470838, 534376, 597033, 682264, 760959, 854755, 941706, 1039255, 1141569, 1254190

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.

Scott R. Shannon, Vertices for n = 2.
Scott R. Shannon, Vertices for n = 3.
Scott R. Shannon, Vertices for n = 4.
Scott R. Shannon, Vertices for n = 6.
Scott R. Shannon, Vertices for n = 8.
Scott R. Shannon, Vertices for n = 10.
Scott R. Shannon, Vertices for n = 14.
Scott R. Shannon, Vertices for n = 15.
Wikipedia, Circular triangle.

Cf. A340685 (regions), A340687 (edges), A340688 (n-gons), A340644, A007678, A092867.

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.

Original entry on oeis.org

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.

A343755 Number of regions formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

Original entry on oeis.org

7, 30, 144, 474, 1324, 2934, 5797, 10614, 17424, 27480, 41845, 61602, 85711, 120120, 159213, 207798, 269668, 349272, 434878, 545496, 661764, 804582, 973471, 1174980, 1374646, 1631304, 1908768, 2218254, 2560198, 2976486, 3378985, 3887796, 4405671, 4995240, 5617689, 6322878

Offset: 1

Scott R. Shannon, Jun 28 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. The number of unbounded areas appears to equal 6*(n^2 - n + 1).

See A344279 for further examples and images of the regions.

			a(1) = 7 as the three connected vertices of a triangle form one polygon along with six outer unbounded areas, seven regions in total.
a(2) = 30 as when the three vertices and three edges points are connected they form twelve polygons, all inside the triangle, along with eighteen outer unbounded areas, thirty regions in total.
a(2) = 144 as when the three vertices and six edges points are connected they form one hundred two polygons, seventy-five inside the triangle and twenty-seven outside, along with forty-two outer unbounded areas, one hundred forty-four regions in total.

Scott R. Shannon, Image for n = 1. In this and other images the triangle's 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 = 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. A344279 (number of polygons), A344657 (number of vertices), A344896 (number of edges), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A121205, A345025.

Conjectured formula: a(n) = A344279(n) + 6*(n^2 - n + 1).

Conjectured formula: a(n) = A344279(n) + A121205(n-1), for n>=7.

A366483 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of vertices in the resulting planar graph.

Original entry on oeis.org

3, 6, 22, 108, 300, 919, 1626, 3558, 5824, 9843, 14352, 23845, 30951, 47196, 62773, 82488, 104544, 144784, 173694, 230008, 276388, 336927, 403452, 509218, 582417, 702228, 824956, 969387, 1098312, 1321978, 1463580, 1724190, 1952509, 2221497, 2505169, 2846908, 3103788, 3556143, 3978763, 4444003

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

nonn

We start with the three corner points of the triangle, and add n further points along each edge. Including the corner points, we end up with n+2 points along each edge, and the edge is divided into n+1 line segments.

Each of the n points added to an edge is joined by 2*n chords to the points that were added to the other two edges. There are 3*n^2 chords.

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 = 10.

Cf. A366484 (interior vertices), A366485 (edges), A366486 (regions).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

a(n) = A366485(n) - A366486(n) + 1 (Euler).

A366484 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of interior vertices in the resulting planar graph.

Original entry on oeis.org

0, 0, 13, 96, 285, 901, 1605, 3534, 5797, 9813, 14319, 23809, 30912, 47154, 62728, 82440, 104493, 144730, 173637, 229948, 276325, 336861, 403383, 509146, 582342, 702150, 824875, 969303, 1098225, 1321888, 1463487, 1724094, 1952410, 2221395, 2505064, 2846800, 3103677, 3556029, 3978646, 4443883

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

nonn

See A366483 for further information.

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 = 10.

Cf. A366483 (vertices), A366485 (edges), A366486 (regions).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

a(n) = A366485(n) - A366486(n) - 3*n - 2 (Euler).

A366486 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of regions in the resulting planar graph.

Original entry on oeis.org

1, 4, 27, 130, 385, 1044, 2005, 4060, 6831, 11272, 16819, 26436, 35737, 52147, 69984, 92080, 117952, 157770, 193465, 249219, 302670, 368506, 443026, 546462, 635125, 757978, 890133, 1041775, 1191442, 1407324, 1581058, 1837417, 2085096, 2365657, 2670429, 3018822, 3328351, 3771595, 4213602

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

nonn

See A366483 for further information.

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 = 10.

Cf. A366483 (vertices), A366484 (interior vertices), A366485 (edges).
If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.

a(n) = A366485(n) - A366483(n) + 1 (Euler).

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.

cp's OEIS Frontend

A340685 The number of regions inside 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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A340686 The number of vertices 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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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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A343755 Number of regions formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

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A366483 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of vertices in the resulting planar graph.

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A366484 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of interior vertices in the resulting planar graph.

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A366486 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of regions in the resulting planar graph.

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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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