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

3,3

Links

Table of n, a(n) for n=3..60.

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.

Example

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.

Crossrefs

Cf. A007678, A062361, A067151, A067152, A067153, A067154, A067155, A067156, A067157, A067158, A067159.

Sequence in context: A217895 A328720 A005128 * A129296 A300837 A321443

Adjacent sequences: A187779 A187780 A187781 * A187783 A187784 A187785

Keyword

nonn,more

Author

Martin Renner, Jan 05 2013

Extensions

a(45)-a(60) from Christopher Scussel, Jun 24 2023

Status

approved