A331932
Triangle read by rows: Take a hexagon with all diagonals drawn, as in A331931. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+4.
Original entry on oeis.org
18, 6, 0, 264, 108, 36, 0, 1344, 654, 252, 12, 6, 4164, 2772, 1020, 228, 24, 0, 10038, 7758, 2424, 516, 72, 24, 0, 21108, 16188, 6060, 1128, 156, 0, 0, 0, 39690, 32022, 13368, 3654, 432, 48, 0, 0, 0, 68052, 56616, 22980, 6084, 888, 120, 12, 0, 0, 0
Offset: 1
A hexagon with no other points along its edges, n = 1, contains 18 triangles, 6 quadrilaterals and no other n-gons, so the first row is [18,6,0]. A hexagon with 1 point dividing its edges, n = 2, contains 264 triangles, 108 quadrilaterals, 36 pentagons and no other n-gons, so the second row is [264,108,36,0].
Triangle begins:
18,6,0
264,108,36,0
1344,654,252,12,6
4164,2772,1020,228,24,0
10038,7758,2424,516,72,24,0
21108,16188,6060,1128,156,0,0,0
39690,32022,13368,3654,432,48,0,0,0
68052,56616,22980,6084,888,120,12,0,0,0
The row sums are A331931.
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.
Original entry on oeis.org
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
- 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.
A330845
The number of edges 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
42, 708, 4020, 15120, 38544, 83136, 169686, 294678, 475500, 746340, 1140624, 1581612, 2296986, 3055734, 3980526, 5391264, 7003662, 8516346, 11094810, 13280970, 16180932, 19971282, 24277212, 28090218, 33683862, 39656604, 45901494, 53121744, 62678268, 69382632
Offset: 1
A330846
The number of vertices 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
19, 301, 1753, 6913, 17713, 38497, 80473, 139927, 225595, 356329, 549967, 756901, 1113283, 1474867, 1913365, 2620789, 3418081, 4119175, 5428993, 6453235, 7861957, 9761335, 11913115, 13695055, 16489633, 19439797, 22464883, 25997329, 30860593, 33866305
Offset: 1
A333075
The number of regions inside an octagon 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
80, 1488, 9312, 31552, 83432, 174816, 339816, 584176, 953416, 1463936, 2173976, 3074784, 4294080, 5790816, 7664880, 9952944, 12757088, 16036096, 20013696, 24577760, 29973528, 36161472, 43314312, 51334672
Offset: 1
A329713
The number of regions inside a heptagon 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
50, 868, 5594, 18396, 48462, 101794, 195714, 336504, 549704, 841890, 1249676, 1774612, 2468572, 3328234, 4414054, 5725034, 7336855, 9233098, 11513419, 14149296, 17254434, 20805554, 24928380, 29573348, 34902155, 40861422, 47613161
Offset: 1
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
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.
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
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.
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
- 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.
A357196
Number of regions in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.
Original entry on oeis.org
1, 7, 25, 55, 97, 151, 217, 295, 385, 475, 601, 715, 865, 1015, 1159, 1351, 1537, 1735, 1945, 2131, 2401, 2647, 2905, 3115, 3457, 3751, 4057, 4357, 4705, 5005, 5401, 5767, 6133, 6535, 6925, 7303, 7777, 8215, 8653, 9025, 9601, 10051, 10585, 11071, 11587, 12151, 12697, 13171, 13825, 14395, 14989
Offset: 0
Showing 1-10 of 17 results.
Comments