A187782 Number of different kinds of polygons in a regular n-gon with all diagonals drawn.
1, 1, 2, 2, 4, 2, 5, 3, 5, 2, 6, 3, 6, 4, 7, 5, 7, 5, 6, 6, 7, 4, 7, 6, 7, 6, 9, 4, 8, 5, 7, 6, 8, 6, 8, 6, 7, 7, 9, 6, 8, 8, 8, 6, 8, 7, 8, 7, 10, 6, 9, 7, 9, 7, 9, 7, 10, 7
Offset: 3
Examples
a(5) = 2 since the 11 regions of the regular pentagon built by all diagonals consist of two different kinds of polygons, i.e., 10 triangles and 1 pentagon. a(6) = 2 since the 24 regions of the regular hexagon built by all diagonals consist of two different kinds of polygons, i.e., 18 triangles and 6 quadrilaterals. a(7) = 4 since the 50 regions of the regular heptagon built by all diagonals consist of four different kinds of polygons, i.e., 35 triangles, 7 quadrilaterals, 7 pentagons and 1 heptagon.
Links
- Sascha Kurz, Anzahl von Dreiecken eines regelmäßigen n-Ecks.
- Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11 (1998), nr. 1, pp. 135-156; doi: 10.1137/S0895480195281246, arXiv: math.MG/9508209.
- Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals.
Crossrefs
Extensions
a(45)-a(60) from Christopher Scussel, Jun 24 2023