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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. 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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. LINKS Table of n, a(n) for n=1..77. Scott R. Shannon, Image of the 6-gon. Scott R. Shannon, Image of the 10-gon. Scott R. Shannon, Image of the 14-gon. Scott R. Shannon, Image of the 24-gon. EXAMPLE 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. CROSSREFS Cf. A351924 (all vertices on diagonal), A352144 (all simple vertices), A292104, A007569, A006561, A146212. Sequence in context: A061106 A352928 A319399 * A161764 A293706 A131055 Adjacent sequences: A352431 A352432 A352433 * A352435 A352436 A352437 KEYWORD nonn AUTHOR Scott R. Shannon, Mar 16 2022 STATUS approved

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Last modified September 9 03:44 EDT 2024. Contains 375759 sequences. (Running on oeis4.)