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.
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.
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.
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.
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.
A333117
The number of vertices inside a pentagram 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
26, 866, 6771, 24221, 64306, 132701, 259761, 453016, 734131, 1134081, 1673056, 2384606, 3326391, 4478286, 5941196, 7710796, 9901136, 12407581, 15497721, 19088991, 23256266, 28021386, 33537586, 39846196, 47092241, 55136771, 64103776, 74213991, 85642556, 98039461
Offset: 1
A333118
The number of edges inside a pentagram 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
65, 1965, 14100, 49760, 130235, 268900, 522770, 911425, 1474680, 2276820, 3354695, 4785575, 6665240, 8973795, 11903415, 15446945, 19825715, 24850460, 31025390, 38221130, 46557865, 56092005, 67123385, 79765335, 94249750, 110346520, 128289075, 148525930, 171374335, 196206590
Offset: 1
A338002
The number of regions inside a 4-pointed star 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
56, 816, 6064, 18152, 52088, 100608, 208168, 336840, 579136, 846560, 1310960, 1784888
Offset: 1
Showing 1-10 of 10 results.
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