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

Author

Sascha Kurz, Jan 06 2002

Keywords

• nonn

Examples

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

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

Crossrefs

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

Formula

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

Extensions

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