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.
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
Keywords
Examples
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.
Links
- 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.
Comments