A187781 Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn.
1, 1, 3, 3, 7, 7, 14, 14, 25, 21, 41, 40, 63, 60, 92, 72, 129, 121, 175, 166, 231, 192, 298, 285, 377, 360, 469, 350, 575, 553, 696, 666, 833, 744, 987, 956, 1159, 1123, 1350, 1165, 1561, 1508, 1793, 1741, 2047, 1875, 2324, 2255, 2625, 2563, 2951, 2761, 3303, 3214, 3682, 3588, 4089, 3695
Offset: 3
Examples
a(5) = 3 since the 11 regions of a regular pentagon with all diagonals drawn consist of three different noncongruent polygons: two different triangles (each 5 times) and 1 pentagon. a(6) = 3 since the 24 regions of the regular hexagon with all diagonals drawn consist of three different noncongruent polygons: 2 triangles (one 6 times, one 12 times) and 1 quadrilateral (6 times). a(7) = 7 since the 50 regions of the regular heptagon with all diagonals drawn consist of seven different noncongruent polygons: 4 triangles (three 7 times, one 14 times), 1 quadrilateral (7 times), 1 pentagon (7 times) 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.
- Index to sequences on drawing diagonals in regular polygons.
Extensions
Corrected a(12) and a(16), extended from a(18) through a(60), corrected small typo in a(7) example - Christopher Scussel, Jun 23 2023