A067151 Number of regions in regular n-gon which are quadrilaterals (4-gons) when all its diagonals are drawn.

0, 0, 6, 7, 24, 36, 90, 132, 168, 234, 378, 600, 672, 901, 954, 1444, 1580, 2520, 2860, 2990, 3696, 4800, 5070, 6750, 7644, 9309, 7920, 12927, 12896, 15576, 16898, 20475, 18684, 25382, 27246, 30966, 32760, 37064, 37170, 45838, 47300, 55350, 60996, 69231, 66864, 80507, 87550, 98124, 103272

Offset

4,3

References

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Links

Scott R. Shannon, Table of n, a(n) for n = 4..765

Sascha Kurz, m-gons in regular n-gons

B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.

B. Poonen and M. Rubinstein, Mathematica programs for these sequences

N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]

Sequences formed by drawing all diagonals in regular polygon

Formula

Conjecture: a(n) ~ c * n^4. Is c = 1/64 ? - Bill McEachen, Mar 03 2024

Example

a(6)=6 because the 6 regions around the center are quadrilaterals.

Crossrefs

Cf. A007678, A067163, A064869, A067152, A067153, A067154, A067155, A067156, A067157, A067158, A067159.

Sequence in context: A282659 A287140 A288771 * A333550 A219916 A135987

Adjacent sequences: A067148 A067149 A067150 * A067152 A067153 A067154

Keyword

nonn

Author

Sascha Kurz, Jan 06 2002

Extensions

Title clarified, a(47) and above by Scott R. Shannon, Dec 04 2021

Status

approved