cp's OEIS Frontend

A332953 The number of regions formed inside an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

%I A332953 #24 May 26 2020 10:25:13
%S A332953 1,5,18,52,125,257,486,832,1333,2027,3048,4304,6057,8167,10749,13929,
%T A332953 18058,22664,28533,34981,42519,51425,62118,73473,86768,101902,118695,
%U A332953 137138,159147,181752,208813,237209,268614,303718,340882,380811,427540,477134,530047
%N A332953 The number of regions formed inside an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.
%C A332953 The terms are from numeric computation - no formula for a(n) is currently known.
%C A332953 Equivalently, this is also the number of regions formed when all the integer points along the x and y axes with 0 <= x <= n and 0 <= y <= n are joined by straight line segments.
%C A332953 If instead one takes points on the x and y axes with coordinates 1, 1/2, 1/3, 1/4, ..., 1/n, 0, and joins them all by line segments, the resulting figure contains only triangles and quadrilaterals, and the number of regions is given by A332358 (and more generally by A332357 if there are m+1 such points on the x axis and n+1 such points on the y axis).
%H A332953 Lars Blomberg, <a href="/A332953/b332953.txt">Table of n, a(n) for n = 1..70</a>
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_3.png">Illustration for n = 2</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_1.png">Illustration for n = 3</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_2.png">Illustration for n = 4</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_4.png">Illustration for n = 5</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_6.png">Illustration for n = 6</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_5.png">Illustration for n = 8</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_7.png">Illustration for n = 10</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_8.png">Illustration for n = 12</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_9.png">Illustration for n = 15</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_10.png">Illustration for n = 5 with random distance-based coloring</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_11.png">Illustration for n = 10 with random distance-based coloring</a>.
%H A332953 Scott R. Shannon, <a href="/A332953/a332953_12.png">Illustration for n = 15 with random distance-based coloring</a>.
%Y A332953 Cf. A333025 (n-gons), A333026 (vertices), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.
%K A332953 nonn
%O A332953 1,2
%A A332953 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 04 2020
%E A332953 a(16) and beyond from _Lars Blomberg_, May 26 2020