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%I #18 Mar 17 2022 08:42:53
%S 0,1,2,2,4,4,6,6,8,8,10,8,12,12,14,14,16,16,18,18,20,20,22,20,24,24,
%T 26,26,28,28,30,30,32,32,34,32,36,36,38,38,40,40,42,42,44,44,46,44,48,
%U 48,50,50,52,52,54,54,56,56,58,56,60,60,62,62,64,64,66,66,68,68,70,68,72,72,74,74,76
%N The number of simple vertices on a diagonal of a regular 2n-gon when all its vertices are connected by lines and where the diagonal passes through the center of the 2n-gon.
%C Excluding a(2), which has its simple vertex at the center of the 4-gon, the terms predominantly follow a pattern of pairs of two equal numbers and where the pair values increment by two. The second term of each pair corresponds to 2n-gons where n is a multiple of 2. These 2n-gons have two vertices that are on the same horizontal line as the central non-simple vertex thus the line joining them will not form a new simple vertex with the central vertical diagonal. Therefore in general a(2*k) = a(2*k-1), k>=1. However this rule is broken when n is a multiple of 12 - for these 2n-gons two of the horizontal lines connecting the left-side and right-side vertices also intersect two non-central diagonals and thus two simple vertices are removed. See the linked image of the 24-gon.
%H Scott R. Shannon, <a href="/A352434/a352434.png">Image of the 6-gon</a>.
%H Scott R. Shannon, <a href="/A352434/a352434_1.png">Image of the 10-gon</a>.
%H Scott R. Shannon, <a href="/A352434/a352434_2.png">Image of the 14-gon</a>.
%H Scott R. Shannon, <a href="/A352434/a352434_3.png">Image of the 24-gon</a>.
%e a(2) = 1 as the 4-gon (square) has one simple vertex at its center when all its vertices are connected by lines.
%e a(3) = 2 as the 6-gon (hexagon) has two simple vertices along the central diagonal when its vertices are connected by lines. See the linked image.
%e a(7) = 6 as the 14-gon has six simple vertices along the central diagonal when its vertices are connected by lines. See the linked image.
%Y Cf. A351924 (all vertices on diagonal), A352144 (all simple vertices), A292104, A007569, A006561, A146212.
%K nonn
%O 1,3
%A _Scott R. Shannon_, Mar 16 2022
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