A344657
Number of vertices formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.
Original entry on oeis.org
3, 10, 85, 310, 999, 2299, 4674, 8878, 14539, 23116, 35922, 53830, 74685, 106957, 140887, 183718, 240108, 315997, 392049, 497518, 599596, 730762, 888903, 1083277, 1257270, 1502830, 1760371, 2047792, 2362620, 2771437, 3129933, 3629443, 4107994, 4670305, 5245447, 5921755
Offset: 1
A344279
Number of polygons formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.
Original entry on oeis.org
1, 12, 102, 396, 1198, 2748, 5539, 10272, 16986, 26934, 41179, 60804, 84769, 119022, 157947, 206352, 268030, 347430, 432820, 543210, 659238, 801804, 970429, 1171662, 1371040, 1627398, 1904550, 2213712, 2555320, 2971260, 3373399, 3881838, 4399329, 4988502, 5610543, 6315312
Offset: 1
A344896
Number of polygon edges formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.
Original entry on oeis.org
3, 21, 186, 705, 2196, 5046, 10212, 19149, 31524, 50049, 77100, 114633, 159453, 225978, 298833, 390069, 508137, 663426, 824868, 1040727, 1258833, 1532565, 1859331, 2254938, 2628309, 3130227, 3664920, 4261503, 4917939, 5742696, 6503331, 7511280, 8507322, 9658806, 10855989, 12237066
Offset: 1
A343755
Number of regions formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.
Original entry on oeis.org
7, 30, 144, 474, 1324, 2934, 5797, 10614, 17424, 27480, 41845, 61602, 85711, 120120, 159213, 207798, 269668, 349272, 434878, 545496, 661764, 804582, 973471, 1174980, 1374646, 1631304, 1908768, 2218254, 2560198, 2976486, 3378985, 3887796, 4405671, 4995240, 5617689, 6322878
Offset: 1
a(1) = 7 as the three connected vertices of a triangle form one polygon along with six outer unbounded areas, seven regions in total.
a(2) = 30 as when the three vertices and three edges points are connected they form twelve polygons, all inside the triangle, along with eighteen outer unbounded areas, thirty regions in total.
a(2) = 144 as when the three vertices and six edges points are connected they form one hundred two polygons, seventy-five inside the triangle and twenty-seven outside, along with forty-two outer unbounded areas, one hundred forty-four regions in total.
- Scott R. Shannon, Image for n = 1. In this and other images the triangle's vertices are highlighted as white dots while the outer open regions are cross-hatched. The key for the edge-number coloring is shown at the top-left of the image. Note the edge count for open areas also includes the two infinite edges
- Scott R. Shannon, Image for n = 2.
- Scott R. Shannon, Image for n = 3.
- Scott R. Shannon, Image for n = 4.
- Scott R. Shannon, Image for n = 5.
- Scott R. Shannon, Image for n = 6.
A386562
Irregular table read by rows: Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: T(n,k) is the number of k-sided finite polygons formed, for k>=3, in the resulting graph.
Original entry on oeis.org
4, 44, 24, 4, 184, 216, 24, 560, 780, 56, 1456, 1844, 224, 12, 3100, 3788, 376, 24, 4, 5860, 7100, 1148, 156, 8, 9860, 12436, 1848, 164, 20, 4, 16044, 19732, 3100, 460, 16, 4, 24744, 29568, 5048, 516, 32, 12, 36780, 43472, 9608, 1400, 68, 20, 52296, 61244, 12628, 1784, 116, 16
Offset: 1
The table begins:
4;
44, 24, 4;
184, 216, 24;
560, 780, 56;
1456, 1844, 224, 12;
3100, 3788, 376, 24, 4;
5860, 7100, 1148, 156, 8;
9860, 12436, 1848, 164, 20, 4;
16044, 19732, 3100, 460, 16, 4;
24744, 29568, 5048, 516, 32, 12;
36780, 43472, 9608, 1400, 68, 20;
52296, 61244, 12628, 1784, 116, 16;
72492, 85672, 20424, 3792, 268, 16;
97812, 115000, 27796, 4820, 344, 24, 8;
129416, 151184, 35716, 6240, 532, 28;
167712, 195816, 46380, 7956, 644, 44;
.
.
- Scott R. Shannon, Image for n = 2. The integer coordinates are highlighted as white dots while the outer open regions, which are not counted, are darkened.
Showing 1-5 of 5 results.
Comments