A331929 -id:A331929 - 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.

A331931 The number of regions inside a hexagon 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

24, 408, 2268, 8208, 20832, 44640, 89214, 154752, 249906, 390012, 590658, 824712, 1183704, 1580868, 2067162, 2770476, 3585582, 4397172, 5665818, 6827736, 8318976, 10209948, 12364098, 14395164, 17194230, 20216808, 23436612, 27124416, 31817676, 35516328

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 01 2020

nonn

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

Scott R. Shannon, Hexagon regions for n = 1.
Scott R. Shannon, Hexagon regions for n = 2.
Scott R. Shannon, Hexagon regions for n = 3.
Scott R. Shannon, Hexagon regions for n = 4.
Scott R. Shannon, Hexagon regions for n = 5.
Scott R. Shannon, Hexagon regions for n = 6.
Scott R. Shannon, Hexagon regions for n = 7.
Scott R. Shannon, Hexagon regions for n = 8.
Scott R. Shannon, Hexagon regions for n = 5, with random distance-based coloring.
Scott R. Shannon, Hexagon regions for n = 6, with random distance-based coloring.
Wikipedia, Hexagon.

Cf. A331932 (n-gons), A330845 (edges), A330846 (vertices), A007678, A092867, A331452, A331929.

a(9)-a(30) from Lars Blomberg, May 12 2020

A329710 The number of edges inside a pentagon 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

20, 290, 2215, 7405, 21150, 43490, 88230, 151135, 250825, 384360, 578840, 814180, 1151525, 1550530, 2063225, 2676925, 3452460, 4333340, 5436210, 6668320, 8154980, 9837690, 11822175, 13993360, 16569650, 19401865, 22636495, 26182350, 30253225, 34608450, 39628050

Offset: 1

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

nonn

See the links in A331929 for images of the pentagons.

Lars Blomberg, Table of n, a(n) for n = 1..35

Cf. A331929 (regions), A331939 (n-gons), A330847 (vertices), A330845, A274586, A332600, A331765.

a(9) and beyond from Lars Blomberg, May 11 2020

A330847 The number of vertices inside a pentagon 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

10, 121, 1055, 3506, 10410, 21111, 43740, 74526, 124490, 190291, 288190, 403321, 573805, 771191, 1027550, 1331896, 1721765, 2157301, 2712175, 3322441, 4067325, 4904491, 5900185, 6975151, 8268755, 9678566, 11297345, 13060231, 15102955, 17263311, 19784995

Offset: 1

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

nonn

See the links in A331929 for images of the pentagons.

Lars Blomberg, Table of n, a(n) for n = 1..35

Cf. A331929 (regions), A331939 (n-gons), A329710 (edges), A330846, A092866, A332599, A007569

a(9) and beyond from Lars Blomberg, May 11 2020

A332421 The number of regions inside a nonagon 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

154, 2754, 16858, 55098, 142318, 298350, 568162, 975294, 1585666, 2426292, 3588508, 5093604, 7067422, 9523746, 12612214, 16351218, 20924029, 26326026, 32789107, 40289238, 49093282, 59181228, 70852528

Offset: 1

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

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

Scott R. Shannon, Nonagon regions for n = 1.
Scott R. Shannon, Nonagon regions for n = 2.
Scott R. Shannon, Nonagon regions for n = 3.
Scott R. Shannon, Nonagon regions with random distance-based coloring for n = 1.
Scott R. Shannon, Nonagon regions with random distance-based coloring for n = 2.
Scott R. Shannon, Nonagon regions with random distance-based coloring for n = 3.
Wikipedia, Nonagon.

Cf. A332427 (n-gons), A332428 (vertices), A332429 (edges), A007678, A092867, A331452, A331929.

a(6)-a(23) from Lars Blomberg, May 16 2020

A333139 The number of regions inside a decagon 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

220, 4220, 25220, 84280, 217800, 456640, 873090, 1501520, 2436020, 3736540, 5523970, 7830800, 10879460, 14665340, 19398660, 25173960, 32203320, 40502280, 50458120, 61995140, 75517160

Offset: 1

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

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

Scott R. Shannon, Decagon regions for n = 1.
Scott R. Shannon, Decagon regions for n = 2.
Scott R. Shannon, Decagon regions for n = 3.
Scott R. Shannon, Decagon regions with random distance-based coloring for n = 1.
Scott R. Shannon, Decagon regions with random distance-based coloring for n = 2.
Scott R. Shannon, Decagon regions with random distance-based coloring for n = 3.
Wikipedia, Decagon.

Cf. A332417 (n-gons), A332418 (vertices), A332419 (edges), A007678, A092867, A331452, A331929.

a(6)-a(21) from Lars Blomberg, May 18 2020

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

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

A335553 a(n) is the number of regions formed by n-secting the angles of a pentagon.

Original entry on oeis.org

1, 10, 11, 60, 131, 90, 276, 310, 326, 510, 706, 580, 981, 1070, 1086, 1390, 1721, 1320, 2136, 2250, 2311, 2730, 3176, 2900, 3746, 3900, 3956, 4530, 5086, 4690, 5821, 5980, 6061, 6760, 7451, 6720, 8286, 8540, 8621, 9420, 10266, 9700, 11241, 11510, 11616, 12560

Offset: 1

Lars Blomberg, Jun 14 2020

nonn

Lars Blomberg, Table of n, a(n) for n = 1..200
Lars Blomberg, Illustration for n=3
Lars Blomberg, Illustration for n=4
Lars Blomberg, Illustration for n=17
Lars Blomberg, Illustration for n=18

Cf. A331929 (n-sected sides, not angles), A335554 (vertices), A335555 (edges), A335556 (ngons).

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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A331931 The number of regions inside a hexagon 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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A329710 The number of edges inside a pentagon 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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A330847 The number of vertices inside a pentagon 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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A332421 The number of regions inside a nonagon 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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A333139 The number of regions inside a decagon 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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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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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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Comments

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Crossrefs

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A335553 a(n) is the number of regions formed by n-secting the angles of a pentagon.

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