A007678 -id:A007678 - cp's OEIS frontend

A331939 Triangle read by rows: Take a pentagon with all diagonals drawn, as in A331929. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+5.

10, 0, 1, 0, 120, 40, 10, 0, 0, 605, 290, 166, 95, 0, 5, 1750, 1420, 550, 150, 30, 0, 0, 4315, 3740, 1920, 640, 95, 20, 5, 6, 9370, 7950, 3610, 1200, 220, 20, 10, 0, 0, 17290, 15705, 7991, 2885, 520, 75, 20, 5, 0, 0, 29590, 28130, 13560, 4320, 860, 150, 0, 0, 0, 0, 0

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 02 2020

See the links in A331929 for images of the pentagons.

			A pentagon with no other points along its edges, n = 1, contains 10 triangles, 1 pentagon and no other n-gons, so the first row is [10,0,1,0]. A pentagon with 1 point dividing its edges, n = 2, contains 120 triangles, 40 quadrilaterals, 10 pentagons and no other n-gons, so the second row is [120, 40, 10, 0, 0].
Triangle begins:
  10,0,1,0
  120,40,10,0,0
  605,290,166,95,0,5
  1750,1420,550,150,30,0,0
  4315,3740,1920,640,95,20,5,6
  9370,7950,3610,1200,220,20,10,0,0
  17290,15705,7991,2885,520,75,20,5,0,0
  29590,28130,13560,4320,860,150,0,0,0,0,0
The row sums are A331929.

Lars Blomberg, Table of n, a(n) for n = 1..735 (the first 35 rows)
Wikipedia, Pentagon.

Cf A331929 (regions), A329710 (edges), A330847 (vertices), A331931, A331906, A007678, A092867, A331452.

A332417 Irregular table read by rows: Take a decagon with all diagonals drawn, as in A333139. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

120, 90, 10, 2040, 1580, 460, 140, 10860, 8570, 4170, 1380, 210, 20, 10, 34360, 30420, 14240, 4020, 1120, 100, 20, 85600, 76920, 38610, 13360, 2650, 550, 110, 176760, 166400, 82560, 24500, 5500, 760, 140, 20, 327550, 320520, 159860, 51610, 10960, 2250, 300, 30, 0, 10

Offset: 1

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

See the links in A333139 for images of the decagons.

			A decagon with no other points along its edges, n = 1, contains 120 triangles, 90 quadrilaterals, 10 pentagons and no other n-gons, so the first row is [120, 90, 10]. A decagon with 1 point dividing its edges, n = 2, contains 2040 triangles, 1580 quadrilaterals, 460 pentagons, 140 hexagons and no other n-gons, so the second row is [2040,1580,460,140].
Table begins:
120, 90, 10;
2040,1580,460,140;
10860,8570,4170,1380,210,20,10;
34360,30420,14240,4020,1120,100,20;
85600,76920,38610,13360,2650,550,110;
176760, 166400, 82560, 24500, 5500, 760, 140, 20;
327550, 320520, 159860, 51610, 10960, 2250, 300, 30, 0, 10;
565060, 549520, 277360, 86540, 18960, 3560, 480, 20, 20;
910920, 891290, 447790, 147300, 32180, 5640, 720, 130, 40, 10;
The row sums are A333139.

Lars Blomberg, Table of n, a(n) for n = 1..181 (the first 21 rows)

Cf. A333139 (regions), A332418 (vertices), A332419 (edges), A007678, A092867, A331452, A331929.

a(29) and beyond from Lars Blomberg, May 18 2020

A332427 Irregular table read by rows: Take a nonagon with all diagonals drawn, as in A332421. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

90, 36, 18, 9, 0, 0, 1, 1332, 918, 414, 90, 6525, 6453, 2529, 1071, 171, 90, 10, 9, 22248, 18882, 10368, 2988, 486, 108, 18, 54558, 50985, 24750, 9387, 2034, 531, 36, 27, 9, 0, 0, 0, 0, 0, 0, 1, 113958, 107676, 54558, 17820, 3672, 612, 36, 18

Offset: 1

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

See the links in A332421 for images of the nonagons.

			A nonagon with no other points along its edges, n = 1, contains 90 triangles, 36 quadrilaterals, 18 pentagons, 9 hexagons, 1 nonagon and no other n-gons, so the first row is [90,36,18,9,0,0,1]. A nonagon with 1 point dividing its edges, n = 2, contains 1332 triangles, 918 quadrilaterals, 414 pentagons, 90 hexagons and no other n-gons, so the second row is [1332,918,414,90].
Table begins:
90,36,18,9,0,0,1;
1332,918,414,90;
6525,6453,2529,1071,171,90,10,9;
22248,18882,10368,2988,486,108,18;
54558,50985,24750,9387,2034,531,36,27,9,0,0,0,0,0,0,1;
113958,107676,54558,17820,3672,612,36,18;
210591,208089,105417,34407,7560,1737,307,45,0,9;
The row sums are A332421.

Lars Blomberg, Table of n, a(n) for n = 1..241 (the first 23 rows)

Cf. A332421 (regions), A332428 (vertices), A332429 (edges), A007678, A092867, A331452, A331929.

a(36) and beyond from Lars Blomberg, May 16 2020

A333076 Irregular table read by rows: Take an octagon with all diagonals drawn, as in A333075. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

56, 24, 800, 608, 64, 16, 4136, 3400, 1272, 464, 40, 13840, 10800, 5296, 1264, 288, 64, 33160, 30048, 14744, 4456, 840, 152, 32, 70832, 62208, 30848, 8656, 1936, 288, 48, 129624, 124224, 61560, 19312, 4168, 840, 64, 16, 0, 8, 225200, 210608, 107552, 32768

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

See the links in A333075 for images of the octagons.

			An octagon with no other points along its edges, n = 1, contains 56 triangles, 24 quadrilaterals and no other n-gons, so the first row is [56,24]. An octagon with 1 point dividing its edges, n = 2, contains 800 triangles, 608 quadrilaterals, 64 pentagons, 16 hexagons and no other n-gons, so the second row is [800,608,64,16].
Table begins:
56,24;
800,608,64,16;
4136,3400,1272,464,40;
13840,10800,5296,1264,288,64;
33160,30048,14744,4456,840,152,32;
70832,62208,30848,8656,1936,288,48;
The rows sums are A333075.

Lars Blomberg, Table of n, a(n) for n = 1..207

Cf. A333075 (regions), A333109 (vertices), A333110 (edges), A331931, A331906, A007678, A092867, A331452.

a(32) and beyond from Lars Blomberg, May 14 2020

A333519 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

Original entry on oeis.org

0, 2, 13, 48, 141, 312, 652, 1160, 1978, 3106, 4775, 6826, 9803, 13328, 17904, 23536, 30652, 38640, 48945, 60300, 74248, 89892, 108768, 128990, 153826, 180206, 211483, 245000, 284375, 325140, 374450, 425312, 484168, 545938, 616981, 690132, 775077, 862220

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 26 2020

nonn

A semicircular polygon with 2n+2 points is created by placing n+2 equally spaced vertices along the semicircle's arc (including the two end vertices). Also place n+2 equally spaced vertices along the diameter (again including the same two end vertices). Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Lars Blomberg, Table of n, a(n) for n = 0..50
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 20.

Cf. A007678, A290865, A255011, A332953, A333282, A334458, A334459.

a(21) and beyond from Lars Blomberg, May 01 2020

A333642 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and three equally spaced points along the diameter (a total of n+3 points). See Comments for precise definition.

Original entry on oeis.org

2, 8, 20, 43, 80, 139, 224, 324, 510, 730, 992, 1373, 1820, 2187, 3040, 3844, 4720, 5916, 7220, 8498, 10472, 12463, 14570, 17278, 20150, 23130, 26964, 30961, 34688, 40265, 45632, 51138, 57970, 65008, 72322, 80979, 89984, 99197, 110240, 121570, 132896, 146818

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 31 2020

nonn

A semicircular polygon with n+3 points is created by placing n+2 equally spaced vertices along the semicircle's arc (including the two end vertices). Also place three equally spaced vertices along the diameter; these are the same two end vertices plus one dividing the diameter. Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Lars Blomberg, Table of n, a(n) for n = 1..100
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 7.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 12.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 17.
Scott R. Shannon, Illustration for n = 19.
Scott R. Shannon, Illustration for n = 20.
Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 19 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 20 with random distance-based coloring.
Wikipedia, Semicircle.

Cf. A330914 (n-gons), A330911 (edges), A330913 (vertices), A333643, A333519, A007678, A290865, A092867, A331452, A331929, A331931.

a(21) and beyond from Lars Blomberg, May 03 2020

A340613 The number of edges on a Reuleaux 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, 42, 264, 957, 2763, 5946, 11976, 20808, 35121, 53853, 81822, 116658, 164409, 222072, 297654, 386613, 499305, 629124, 789156, 970404, 1189923, 1435689, 1726980, 2050254, 2428101, 2843862, 3323646, 3847863, 4446861, 5096730, 5833128, 6627840, 7520769, 8480367, 9550638, 10695942, 11966043

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 14 2021

nonn

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

See A340639 for images of the regions and A340644 for images of the vertices.

Wikipedia, Reuleaux triangle.

Cf. A340639 (regions), A340644 (vertices), A340614 (n-gons), A007678, A092867.

A340614 Irregular table read by rows: Take a Reuleaux triangle with all diagonals drawn, as in A340639. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

Original entry on oeis.org

1, 18, 6, 79, 51, 12, 3, 252, 192, 60, 6, 6, 576, 600, 168, 73, 15, 1170, 1380, 390, 126, 6, 12, 2248, 2589, 894, 288, 66, 18, 3, 4026, 4332, 1662, 480, 108, 30, 6426, 7182, 2988, 943, 189, 36, 9942, 11268, 4470, 1266, 300, 84, 0, 6, 14508, 16941, 7098, 2119, 435, 120, 6, 6

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 14 2021

See A340639 for images of the regions and A340644 for images of the vertices.

			A Reuleaux triangle with 1 point dividing its edges, n = 2, contains 18 triangles, 6 quadrilaterals and no other n-gons, so the second row is [18, 6]. A Reuleaux triangle with 2 points dividing its edges, n = 3, contains 79 triangles, 51 quadrilaterals, 12 pentagons, 3 hexagons and no other n-gons, so the third row is [79, 51, 12, 3].
The table begins:
1;
18, 6;
79, 51, 12, 3;
252, 192, 60, 6, 6;
576, 600, 168, 73, 15;
1170, 1380, 390, 126, 6, 12;
2248, 2589, 894, 288, 66, 18, 3;
4026, 4332, 1662, 480, 108, 30;
6426, 7182, 2988, 943, 189, 36;
9942, 11268, 4470, 1266, 300, 84, 0, 6;
14508, 16941, 7098, 2119, 435, 120, 6, 6;
21234, 23598, 10194, 3090, 636, 132, 12, 6;
28881, 34086, 13935, 4528, 1041, 177, 24, 3;
39588, 45384, 19470, 5796, 1380, 198, 48;
52197, 60744, 26409, 7831, 1914, 366, 21, 15;
68646, 78492, 33810, 10668, 2358, 396, 60;
87642, 100701, 44670, 13942, 2931, 555, 66, 21, 6;
111084, 127290, 55818, 17082, 3912, 696, 132, 6;
138453, 158907, 70158, 22233, 4869, 1002, 87, 15, 0, 4;
171276, 194622, 87312, 26748, 6132, 846, 174, 6, 6;

Wikipedia, Reuleaux triangle.

Cf. A340639 (regions), A340644 (vertices), A340613 (edges), A007678, A092867.

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.

Original entry on oeis.org

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.

cp's OEIS Frontend

A331939 Triangle read by rows: Take a pentagon with all diagonals drawn, as in A331929. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+5.

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A332417 Irregular table read by rows: Take a decagon with all diagonals drawn, as in A333139. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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A332427 Irregular table read by rows: Take a nonagon with all diagonals drawn, as in A332421. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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A333076 Irregular table read by rows: Take an octagon with all diagonals drawn, as in A333075. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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A333519 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

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A333642 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and three equally spaced points along the diameter (a total of n+3 points). See Comments for precise definition.

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A340613 The number of edges on a Reuleaux 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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A340614 Irregular table read by rows: Take a Reuleaux triangle with all diagonals drawn, as in A340639. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

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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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