cp's OEIS Frontend

A333642 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and three equally spaced points along the diameter (a total of n+3 points). See Comments for precise definition.

%I A333642 #26 Jun 02 2025 22:32:13
%S A333642 2,8,20,43,80,139,224,324,510,730,992,1373,1820,2187,3040,3844,4720,
%T A333642 5916,7220,8498,10472,12463,14570,17278,20150,23130,26964,30961,34688,
%U A333642 40265,45632,51138,57970,65008,72322,80979,89984,99197,110240,121570,132896,146818
%N A333642 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and three equally spaced points along the diameter  (a total of n+3 points). See Comments for precise definition.
%C A333642 A semicircular polygon with n+3 points is created by placing n+2 equally spaced vertices along the semicircle's arc (including the two end vertices). Also place three equally spaced vertices along the diameter; these are the same two end vertices plus one dividing the diameter. Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.
%H A333642 Lars Blomberg, <a href="/A333642/b333642.txt">Table of n, a(n) for n = 1..100</a>
%H A333642 Scott R. Shannon, <a href="/A333642/a333642.png">Illustration for n = 2</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_1.png">Illustration for n = 3</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_2.png">Illustration for n = 4</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_3.png">Illustration for n = 5</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_4.png">Illustration for n = 7</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_5.png">Illustration for n = 10</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_6.png">Illustration for n = 12</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_7.png">Illustration for n = 15</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_8.png">Illustration for n = 17</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_10.png">Illustration for n = 19</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_9.png">Illustration for n = 20</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_13.png">Illustration for n = 10 with random distance-based coloring</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_14.png">Illustration for n = 15 with random distance-based coloring</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_11.png">Illustration for n = 19 with random distance-based coloring</a>.
%H A333642 Scott R. Shannon, <a href="/A333642/a333642_12.png">Illustration for n = 20 with random distance-based coloring</a>.
%H A333642 Wikipedia, <a href="https://en.wikipedia.org/wiki/Semicircle">Semicircle</a>.
%Y A333642 Cf. A330914 (n-gons), A330911 (edges), A330913 (vertices), A333643, A333519, A007678, A290865, A092867, A331452, A331929, A331931.
%K A333642 nonn
%O A333642 1,1
%A A333642 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 31 2020
%E A333642 a(21) and beyond from _Lars Blomberg_, May 03 2020