

A352434


The number of simple vertices on a diagonal of a regular 2ngon when all its vertices are connected by lines and where the diagonal passes through the center of the 2ngon.


1



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, 26, 26, 28, 28, 30, 30, 32, 32, 34, 32, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 44, 48, 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
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OFFSET

1,3


COMMENTS

Excluding a(2), which has its simple vertex at the center of the 4gon, 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 2ngons where n is a multiple of 2. These 2ngons have two vertices that are on the same horizontal line as the central nonsimple 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*k1), k>=1. However this rule is broken when n is a multiple of 12  for these 2ngons two of the horizontal lines connecting the leftside and rightside vertices also intersect two noncentral diagonals and thus two simple vertices are removed. See the linked image of the 24gon.


LINKS



EXAMPLE

a(2) = 1 as the 4gon (square) has one simple vertex at its center when all its vertices are connected by lines.
a(3) = 2 as the 6gon (hexagon) has two simple vertices along the central diagonal when its vertices are connected by lines. See the linked image.
a(7) = 6 as the 14gon has six simple vertices along the central diagonal when its vertices are connected by lines. See the linked image.


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



