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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A352434 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.

Original entry on oeis.org

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
Offset: 1

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Author

Scott R. Shannon, Mar 16 2022

Keywords

Comments

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.

Examples

			a(2) = 1 as the 4-gon (square) has one simple vertex at its center when all its vertices are connected by lines.
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.
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.

Links

Crossrefs

Cf. A351924 (all vertices on diagonal), A352144 (all simple vertices), A292104, A007569, A006561, A146212.
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