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A333519 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

%I A333519 #35 Jun 02 2025 22:31:38
%S A333519 0,2,13,48,141,312,652,1160,1978,3106,4775,6826,9803,13328,17904,
%T A333519 23536,30652,38640,48945,60300,74248,89892,108768,128990,153826,
%U A333519 180206,211483,245000,284375,325140,374450,425312,484168,545938,616981,690132,775077,862220
%N A333519 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.
%C A333519 A semicircular polygon with 2n+2 points is created by placing n+2 equally spaced vertices along the semicircle's arc (including the two end vertices). Also place n+2 equally spaced vertices along the diameter (again including the same two end vertices). Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.
%H A333519 Lars Blomberg, <a href="/A333519/b333519.txt">Table of n, a(n) for n = 0..50</a>
%H A333519 Scott R. Shannon, <a href="/A333519/a333519.png">Illustration for n = 2</a>.
%H A333519 Scott R. Shannon, <a href="/A333519/a333519_1.png">Illustration for n = 3</a>.
%H A333519 Scott R. Shannon, <a href="/A333519/a333519_2.png">Illustration for n = 5</a>.
%H A333519 Scott R. Shannon, <a href="/A333519/a333519_3.png">Illustration for n = 10</a>.
%H A333519 Scott R. Shannon, <a href="/A333519/a333519_4.png">Illustration for n = 15</a>.
%H A333519 Scott R. Shannon, <a href="/A333519/a333519_5.png">Illustration for n = 20</a>.
%Y A333519 Cf. A007678, A290865, A255011, A332953, A333282, A334458, A334459.
%K A333519 nonn
%O A333519 0,2
%A A333519 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 26 2020
%E A333519 a(21) and beyond from _Lars Blomberg_, May 01 2020