A335646
Irregular table read by rows: row n gives the number of 5-gon to k-gon contacts, with k>=5, for a regular n-gon with all diagonals drawn.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 39, 42, 15, 0, 15, 0, 102, 18, 190, 38, 19, 20, 80, 273, 210, 21, 154, 44, 529, 322, 69, 144, 750, 350, 598, 156, 26, 1215, 432, 81, 560, 56, 928, 406, 29, 0, 0, 0, 29, 300, 60, 2139, 248, 93, 1568, 704, 64, 1782, 792, 132
Offset: 3
The table begins:
.
0;
0;
0;
0;
0;
0;
0;
0;
11;
0;
39;
42;
15,0,15;
0;
102;
18;
190,38,19;
20,80;
273,210,21;
154,44;
529,322,69;
144;
750,350;
598,156,26;
1215,432,81;
560,56;
928,406,29,0,0,0,29;
300,60;
2139,248,93;
1568,704,64;
1782,792,132;
A337330
Irregular table read by rows: row n gives the number of 6-gon to k-gon contacts, with k>=6, for a regular n-gon with all diagonals drawn, with n>=25.
Original entry on oeis.org
50, 0, 108, 0, 0, 0, 124, 32, 66, 136, 70, 144, 148, 76, 390, 120, 328, 82, 42, 86, 86, 0, 540, 92, 92, 188, 94, 94, 0, 196, 98, 750, 100, 816, 416, 104, 1272, 432, 220, 110, 728, 570, 570, 406, 348, 116, 1062, 354, 300, 854, 366, 122, 1488, 124, 1512, 252, 126, 576, 2080, 130, 260, 2112
Offset: 25
The table begins:
.
50;
0;
108;
0;
0;
0;
124;
32;
66;
136;
70;
144;
148;
76;
390;
120;
328, 82;
42;
A340639
The number of regions inside 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
1, 24, 145, 516, 1432, 3084, 6106, 10638, 17764, 27336, 41233, 58902, 82675, 111864, 149497, 194430, 250534, 316020, 395728, 487122, 596434, 720162, 865321, 1027974, 1216291, 1425348, 1664539, 1928022, 2226658, 2553204, 2920378, 3319536, 3764848, 4246638, 4780489, 5355414, 5988973
Offset: 1
- 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 = 5.
- Scott R. Shannon, Regions for n = 6.
- Scott R. Shannon, Regions for n = 10.
- Scott R. Shannon, Regions for n = 11.
- Scott R. Shannon, Regions for n = 9 with random distance-based coloring.
- Scott R. Shannon, Regions for n = 10 with random distance-based coloring.
- Wikipedia, Reuleaux triangle.
A340644
The number of vertices 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, 19, 120, 442, 1332, 2863, 5871, 10171, 17358, 26518, 40590, 57757, 81735, 110209, 148158, 192184, 248772, 313105, 393429, 483283, 593490, 715528, 861660, 1022281, 1211811, 1418515, 1659108, 1919842, 2220204, 2543527, 2912751, 3308305, 3755922, 4233730, 4770150, 5340529, 5977071
Offset: 1
A342236
a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, as in A342222, but for odd m the central m-sided polygon is not considered. Otherwise a(n) = -1 if no such m exists.
Original entry on oeis.org
4, 6, 7, 9, 15, 13, 35, 29, 29, 40, 93, 43, 399, 212
Offset: 3
See also
A341729 and
A341730 for the maximum number of sides in any cell.
A329714
Irregular table read by rows: Take a heptagon with all diagonals drawn, as in A329713. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.
Original entry on oeis.org
35, 7, 7, 0, 1, 504, 224, 112, 28, 2331, 1883, 1008, 273, 92, 7, 7658, 6314, 3416, 798, 182, 28, 18662, 17514, 8463, 2898, 714, 175, 28, 7, 0, 0, 0, 1, 40404, 35462, 18508, 5796, 1330, 266, 28
Offset: 1
A heptagon with no other points along its edges, n = 1, contains 35 triangles, 7 quadrilaterals, 7 pentagons, 1 heptagon and no other n-gons, so the first row is [35,7,7,0,1]. A heptagon with 1 point dividing its edges, n = 2, contains 504 triangles, 224 quadrilaterals, 112 pentagons, 28 hexagons and no other n-gons, so the second row is [504,224,112,28].
Triangle begins:
35, 7, 7, 0, 1;
504, 224, 112, 28;
2331, 1883, 1008, 273, 92, 7;
7658, 6314, 3416, 798, 182, 28;
18662, 17514, 8463, 2898, 714, 175, 28, 7, 0, 0, 0, 1;
40404, 35462, 18508, 5796, 1330, 266, 28;
73248, 71596, 35777, 11669, 2654, 651, 70, 49;
The row sums are A329713.
A331455
Number of regions in a "cross" of width 3 and height n (see Comments for definition).
Original entry on oeis.org
64, 104, 176, 304, 492, 778, 1176, 1732, 2446, 3416, 4614, 6172, 8060, 10340, 13052, 16388, 20228, 24852, 30134, 36206, 43076, 51092, 60010, 70186, 81498, 94180, 108140, 123938, 141074, 160308, 181320, 204328, 229288, 256574, 285856, 318124, 352838, 390338
Offset: 2
- Lars Blomberg, Table of n, a(n) for n = 2..50
- Scott R. Shannon, Illustration for cross of height 2.
- Scott R. Shannon, Illustration for cross of height 3.
- Scott R. Shannon, Illustration for cross of height 4.
- Scott R. Shannon, Illustration for cross of height 5.
- Scott R. Shannon, Illustration for cross of height 6.
- Scott R. Shannon, Illustration for cross of height 9.
- Scott R. Shannon, Illustration for cross of height 3 using random distance-based coloring.
- Scott R. Shannon, Illustration for cross of height 4 using random distance-based coloring.
- Scott R. Shannon, Illustration for cross of height 5 using random distance-based coloring.
- Scott R. Shannon, Illustration for cross of height 6 using random distance-based coloring.
- Scott R. Shannon, Illustration for cross of height 7 using random distance-based coloring.
- Scott R. Shannon, Colored illustration for a different-shaped cross, with arms of lengths 2,2,4. (There are 21858 regions.)
- N. J. A. Sloane, Illustration for cross of height 2.
- N. J. A. Sloane, Illustration for cross of height 3. (One of the "arms" has been cropped by the scanner, but all four arms are the same.)
- N. J. A. Sloane, Illustration for cross of height 4.
- N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
See
A331456 for crosses in which the arms have equal length.
A331452 is a similar sequence for a rectangular region;
A007678 for a polygonal region.
A331907
Triangle read by rows: Take a pentagram with all diagonals drawn, as in A331906. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+2.
Original entry on oeis.org
40, 0, 0, 590, 420, 80, 10, 2890, 3030, 1130, 230, 50, 9540, 10530, 4290, 980, 190, 10, 22730, 28390, 10960, 3200, 550, 80, 20, 47610, 57450, 23270, 6530, 1160, 160, 20, 0, 90080, 109160, 47430, 13430, 2460, 410, 40, 0, 0, 154840, 193480, 82330, 22410, 4620
Offset: 1
A pentagram with no other points along its edges, n = 1, contains 40 triangles and no other n-gons, so the first row is [40,0,0]. A pentagram with 1 point dividing its edges, n = 2, contains 590 triangles, 420 quadrilaterals, 80 pentagons and 10 hexagons, so the second row is [590,420,80,10].
Triangle begins:
40,0,0
590, 420, 80, 10
2890, 3030, 1130, 230, 50
9540, 10530, 4290, 980, 190, 10
22730, 28390, 10960, 3200, 550, 80, 20
47610, 57450, 23270, 6530, 1160, 160, 20, 0
The row sums are A331906.
A331908
The number of regions inside a hexagram 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
168, 3588, 20424, 73860, 189468, 402216, 782808, 1385040, 2214108, 3423840, 5196312, 7218552, 10353432, 13823772, 18047124, 24083736, 31051152, 38334972, 48877440, 59201544, 72052956, 88004184, 106601088, 124009020
Offset: 1
A331909
Triangle read by rows: Take a hexagram with all diagonals drawn, as in A331908. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n+5.
Original entry on oeis.org
132, 36, 0, 0, 2052, 1188, 324, 24, 0, 10440, 7956, 1728, 300, 0, 0, 33672, 28812, 9276, 1836, 228, 24, 12, 83040, 75276, 24948, 5436, 708, 60, 0, 0, 172140, 162060, 54732, 11280, 1836, 168, 0, 0, 0, 322284, 315492, 114624, 25980, 3948, 456, 24, 0, 0, 0
Offset: 1
A hexagram with no other points along its edges, n = 1, contains 132 triangles, 36 quadrilaterals and no other n-gons, so the first row is [132,36,0,0]. A hexagram with 1 point dividing its edges, n = 2, contains 2052 triangles, 1188 quadrilaterals, 324 pentagons, 24 hexagons and no other n-gons, so the second row is [2052,1188,324,24,0].
Triangle begins:
132, 36, 0, 0
2052, 1188, 324, 24, 0
10440, 7956, 1728, 300, 0, 0
33672, 28812, 9276, 1836, 228, 24, 12
83040, 75276, 24948, 5436, 708, 60, 0, 0
The row sums are A331908.
Comments