A067159 -id:A067159 - cp's OEIS frontend

A067151 Number of regions in regular n-gon which are quadrilaterals (4-gons) when all its diagonals are drawn.

0, 0, 6, 7, 24, 36, 90, 132, 168, 234, 378, 600, 672, 901, 954, 1444, 1580, 2520, 2860, 2990, 3696, 4800, 5070, 6750, 7644, 9309, 7920, 12927, 12896, 15576, 16898, 20475, 18684, 25382, 27246, 30966, 32760, 37064, 37170, 45838, 47300, 55350, 60996, 69231, 66864, 80507, 87550, 98124, 103272

Offset: 4

Sascha Kurz, Jan 06 2002

nonn

			a(6)=6 because the 6 regions around the center are quadrilaterals.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 4..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Mathematica programs for these sequences
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067163, A064869, A067152, A067153, A067154, A067155, A067156, A067157, A067158, A067159.

Conjecture: a(n) ~ c * n^4. Is c = 1/64 ? - Bill McEachen, Mar 03 2024

Title clarified, a(47) and above by Scott R. Shannon, Dec 04 2021

A067152 Number of pentagonal regions in regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 0, 7, 0, 18, 10, 44, 0, 117, 98, 150, 128, 357, 72, 646, 580, 903, 814, 1564, 840, 2050, 2106, 2862, 2128, 3625, 1440, 5146, 4896, 6105, 5542, 8190, 7452, 10471, 10184, 14235, 13160, 16564, 11382, 21156, 20548, 24300, 23920, 30362, 26112, 35231, 32700, 40341, 38532, 51834, 42012, 58905

Offset: 5

Sascha Kurz, Jan 06 2002

nonn

			a(5) = 1 because only the center-region is a pentagon.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 5..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067164, A064869, A067151, A067153, A067154, A067155, A067156, A067157, A067158, A067159.

a(49) and beyond from Scott R. Shannon, Dec 04 2021

Definition clarified by N. J. A. Sloane, Jun 09 2025

A067153 Number of hexagonal regions in regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 0, 9, 0, 22, 0, 39, 0, 105, 48, 136, 18, 190, 120, 462, 66, 644, 72, 875, 390, 1296, 952, 1595, 450, 1891, 1472, 3201, 2346, 3640, 2124, 4773, 2698, 5577, 4000, 7298, 3444, 7912, 6336, 10980, 6532, 10904, 7824, 14651, 12150, 16779, 13260, 20299, 13176, 21560, 18200, 26961, 21634, 29500

Offset: 6

Sascha Kurz, Jan 06 2002

nonn

			a(9)=9 because drawing the regular 9-gon with all its diagonals yields 9 hexagons.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 6..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067165, A064869, A067151, A067152, A067154, A067155, A067156, A067157, A067158, A067159.

a(54) and beyond from Scott R. Shannon, Dec 04 2021

Definition clarified by N. J. A. Sloane, Jun 09 2025

A067154 Number of heptagonal regions in regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 15, 0, 17, 18, 57, 0, 21, 44, 115, 0, 150, 104, 81, 112, 116, 0, 155, 224, 429, 306, 560, 180, 555, 836, 663, 640, 1025, 378, 1419, 660, 1710, 1564, 1786, 1200, 2352, 1050, 2754, 2236, 2597, 2700, 3410, 2240, 3078, 3190, 4602, 1860, 5551, 4898, 6363, 5056, 8515, 4950

Offset: 7

Sascha Kurz, Jan 06 2002

nonn

			a(7)=1 because the center-region is a heptagon.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 7..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067166, A064869, A067151, A067152, A067153, A067155, A067156, A067157, A067158, A067159.

a(62) and beyond from Scott R. Shannon, Dec 04 2021

Definition clarified by N. J. A. Sloane, Jun 09 2025

A067155 Number of octagonal regions in regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 0, 0, 0, 13, 0, 0, 0, 34, 0, 38, 20, 0, 44, 23, 0, 50, 26, 108, 28, 145, 0, 217, 0, 264, 102, 315, 72, 407, 190, 546, 200, 656, 42, 903, 528, 810, 598, 1175, 288, 1078, 550, 1479, 780, 1166, 486, 1705, 784, 2451, 1276, 3068, 960, 3172, 1860, 4347, 2432, 4225, 2376, 4958, 2992, 3519, 2380

Offset: 8

Sascha Kurz, Jan 06 2002

nonn

			a(13)=13 because drawing the regular 13-gon and all its diagonals yields 13 octagons.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 8..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067167, A064869, A067151, A067152, A067153, A067154, A067156, A067157, A067158, A067159.

a(65) and beyond from Scott R. Shannon, Dec 04 2021

Definition clarified by Hugo Pfoertner, Dec 04 2021

A067156 Number of regions in regular n-gon which are 9-gons.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 123, 0, 0, 88, 45, 0, 0, 0, 0, 0, 51, 0, 0, 0, 165, 0, 114, 0, 118, 120, 61, 124, 0, 192, 195, 66, 67, 272, 138, 0, 568, 360, 146, 222, 600, 0, 231, 156, 237, 800, 567, 410, 664, 84, 255, 344, 174

Offset: 9

Sascha Kurz, Jan 06 2002

nonn

			a(9)=1 because drawing the regular 9-gon with all its diagonals yields 1 9-gon.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 9..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067168, A064869, A067151, A067152, A067153, A067154, A067155, A067157, A067158, A067159.

a(86) and beyond by Scott R. Shannon, Dec 04 2021

A067157 Number of regions in regular n-gon which are 10-gons.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 87, 0, 62, 0, 0, 0, 0, 0, 74, 0, 0, 0, 41, 0, 0, 44, 0, 0, 235, 48, 147, 100, 51, 0, 159, 54, 110, 56, 114, 58, 177, 0, 183, 62, 378, 256, 195, 0, 134, 136, 621, 210, 71, 144, 438, 222, 750, 76, 385, 78, 1185, 80, 648, 82, 830, 336, 935

Offset: 10

Sascha Kurz, Jan 06 2002

nonn

			a(29)=87 because drawing the regular 29-gon with all its diagonals yields 87 10-gons.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 10..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067169, A064869, A067151, A067152, A067153, A067154, A067155, A067156, A067158, A067159.

a(83) and beyond by Scott R. Shannon, Dec 04 2021

A067158 Number of regions in regular n-gon which are 11-gons.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 71, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 99, 0, 0, 102, 103, 0, 0, 0, 0, 108, 0

Offset: 11

Sascha Kurz, Jan 06 2002

nonn

			a(11)=1 because drawing the regular 11-gon with all its diagonals yields 1 11-gon.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 11..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A064869, A067151, A067152, A067153, A067154, A067155, A067156, A067157, A067159.

a(110) and beyond by Scott R. Shannon, Dec 04 2021

A187782 Number of different kinds of polygons in a regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 5, 3, 5, 2, 6, 3, 6, 4, 7, 5, 7, 5, 6, 6, 7, 4, 7, 6, 7, 6, 9, 4, 8, 5, 7, 6, 8, 6, 8, 6, 7, 7, 9, 6, 8, 8, 8, 6, 8, 7, 8, 7, 10, 6, 9, 7, 9, 7, 9, 7, 10, 7

Offset: 3

Martin Renner, Jan 05 2013

			a(5) = 2 since the 11 regions of the regular pentagon built by all diagonals consist of two different kinds of polygons, i.e., 10 triangles and 1 pentagon.
a(6) = 2 since the 24 regions of the regular hexagon built by all diagonals consist of two different kinds of polygons, i.e., 18 triangles and 6 quadrilaterals.
a(7) = 4 since the 50 regions of the regular heptagon built by all diagonals consist of four different kinds of polygons, i.e., 35 triangles, 7 quadrilaterals, 7 pentagons and 1 heptagon.

Sascha Kurz, Anzahl von Dreiecken eines regelmäßigen n-Ecks.
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11 (1998), nr. 1, pp. 135-156; doi: 10.1137/S0895480195281246, arXiv: math.MG/9508209.
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals.

Cf. A007678, A062361, A067151, A067152, A067153, A067154, A067155, A067156, A067157, A067158, A067159.

a(45)-a(60) from Christopher Scussel, Jun 24 2023

cp's OEIS Frontend

A067151 Number of regions in regular n-gon which are quadrilaterals (4-gons) when all its diagonals are drawn.

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A067152 Number of pentagonal regions in regular n-gon with all diagonals drawn.

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A067153 Number of hexagonal regions in regular n-gon with all diagonals drawn.

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A067154 Number of heptagonal regions in regular n-gon with all diagonals drawn.

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A067155 Number of octagonal regions in regular n-gon with all diagonals drawn.

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A067156 Number of regions in regular n-gon which are 9-gons.

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A067157 Number of regions in regular n-gon which are 10-gons.

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A067158 Number of regions in regular n-gon which are 11-gons.

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A187782 Number of different kinds of polygons in a regular n-gon with all diagonals drawn.

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