A330847 -id:A330847 - cp's OEIS frontend

A331929 The number of regions inside a pentagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

11, 170, 1161, 3900, 10741, 22380, 44491, 76610, 126336, 194070, 290651, 410860, 577721, 779340, 1035676, 1345030, 1730696, 2176040, 2724036, 3345880, 4087656, 4933200, 5921991, 7018210, 8300896, 9723300, 11339151, 13122120, 15150271, 17345140, 19843056

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.

Lars Blomberg, Table of n, a(n) for n = 1..35
Scott R. Shannon, Pentagon regions for n = 1.
Scott R. Shannon, Pentagon regions for n = 2.
Scott R. Shannon, Pentagon regions for n = 3.
Scott R. Shannon, Pentagon regions for n = 4.
Scott R. Shannon, Pentagon regions for n = 5.
Scott R. Shannon, Pentagon regions for n = 6.
Scott R. Shannon, Pentagon regions for n = 7.
Scott R. Shannon, Pentagon regions for n = 8.
Scott R. Shannon, Pentagon regions for n = 5, random distance-based coloring.
Scott R. Shannon, Pentagon regions for n = 6, random distance-based coloring
Wikipedia, Pentagon.

Cf. A331939 (n-gons), A329710 (edges), A330847 (vertices), A007678, A092867, A331452, A331931.

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

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.

Original entry on oeis.org

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.

A335554 a(n) is the number of vertices formed by n-secting the angles of a pentagon.

Original entry on oeis.org

5, 11, 10, 46, 135, 71, 280, 266, 325, 456, 710, 506, 985, 981, 1085, 1291, 1725, 1091, 2140, 2121, 2310, 2586, 3180, 2706, 3750, 3726, 3955, 4341, 5090, 4436, 5825, 5761, 6060, 6531, 7455, 6241, 8290, 8276, 8620, 9146, 10270, 9326, 11245, 11206, 11615, 12241

Offset: 1

Lars Blomberg, Jun 20 2020

nonn

See A335553 for illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..200

Cf. A330847 (n-sected sides, not angles), A335553 (regions), A335555 (edges), A335556 (ngons).

A367322 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of vertices in the resulting planar graph.

Original entry on oeis.org

3, 10, 5, 58, 37, 10, 178, 257, 121, 19, 558, 817, 1055, 301, 42, 1255, 2757, 3506, 1753, 708, 57, 2532, 4825, 10410, 6913, 5369, 1145, 135, 4786, 12293, 21111, 17713, 17417, 8417, 2395, 171, 7804, 19241, 43740, 38497, 47796, 29121, 16434, 3581, 341

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Nov 14 2023

See A367323 and the cross references for further images of the n-gons.

			The table begins:
3, 10, 58, 178, 558, 1255, 2532, 4786, 7804, 12292, 18966, 28540, 39117, 56107, ...
5, 37, 257, 817, 2757, 4825, 12293, 19241, 33549, 49577, 87685, 101981, 178465, ...
10, 121, 1055, 3506, 10410, 21111, 43740, 74526, 124490, 190291, 288190, ...
19, 301, 1753, 6913, 17713, 38497, 80473, 139927, 225595, 356329, 549967, ...
42, 708, 5369, 17417, 47796, 99261, 194278, 331955, 546805, 833946, 1245314, ...
57, 1145, 8417, 29121, 80345, 167105, 333297, 570969, 939113, 1441153, 2153937, ...
135, 2395, 16434, 53155, 141147, 293374, 565767, 966493, 1580940, 2411533, ...
171, 3581, 23651, 80191, 213041, 444251, 862711, 1481141, 2413721, 3701951, ...
341, 6062, 39248, 126061, 329131, 684223, 1307845, 2233815, 3639020, 5549952, ...
313, 7513, 47293, 167941, 450457, 931345, 1830625, 3132349, 5103589, 7825201, ...
728, 12845, 79859, 255711, 660140, 1372008, 2608476, 4454477, 7236853, ...
771, 16871, 103517, 343855, 881959, 1847525, 3504971, 6013953, 9739227, ...
1380, 24136, 145635, 465721, 1192710, 2478121, 4694040, 8014891, 12995535, ...
1393, 30305, 182785, 602337, 1533681, 3211873, 6067041, 10402769, 16824161, ...
2397, 41583, 245684, 783803, 1995341, 4145230, 7829333, 13366897, ...
1855, 46801, 291637, 967123, 2476873, 5166055, 9798175, ...
3895, 67090, 389690, 1241765, 3146533, 6535659, 12317567, ...
3861, 80921, 466141, 1522001, 3840181, 8027441, 15094521, ...
6006, 102817, 589029, 1875511, 4734786, 9833104, ...
5963, 122849, 694387, 2259005, 5671887, 11852105, ...
8878, 151180, 856589, 2725041, 6858600, 14242153, ...
7321, 169297, 979777, 3205921, 8068321, ...
12675, 214851, 1206050, 3834451, 9626475, ...
.
.
.

Scott R. Shannon, Image for T(12,2).
Scott R. Shannon, Image for T(23,1).

Cf. A367323 (regions), A367324 (edges), A274585 (1st row), A331449 (2nd row), A330847 (3rd row), A330846 (4th row), A333113 (5th row), A333109 (6th row), A332428 (7th row), A332418 (8th row), A007569 (1st column).

T(n,k) = A367324(n,k) - A367323(n,k) + 1 (Euler).

cp's OEIS Frontend

A331929 The number of regions 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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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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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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A335554 a(n) is the number of vertices formed by n-secting the angles of a pentagon.

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A367322 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of vertices in the resulting planar graph.

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