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

%I A067151 #38 Jun 08 2025 13:51:31
%S A067151 0,0,6,7,24,36,90,132,168,234,378,600,672,901,954,1444,1580,2520,2860,
%T A067151 2990,3696,4800,5070,6750,7644,9309,7920,12927,12896,15576,16898,
%U A067151 20475,18684,25382,27246,30966,32760,37064,37170,45838,47300,55350,60996,69231,66864,80507,87550,98124,103272
%N A067151 Number of regions in regular n-gon which are quadrilaterals (4-gons) when all its diagonals are drawn.
%D A067151 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.
%H A067151 Scott R. Shannon, <a href="/A067151/b067151.txt">Table of n, a(n) for n = 4..765</a>
%H A067151 Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/drawing/drawing.html">m-gons in regular n-gons</a>
%H A067151 B. Poonen and M. Rubinstein, <a href="https://arxiv.org/abs/math/9508209">The number of intersection points made by the diagonals of a regular polygon</a>, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.
%H A067151 B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.m">Mathematica programs for these sequences</a>
%H A067151 N. J. A. Sloane, <a href="/A331450/a331450.jpg">Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19.</a> [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).]
%H A067151 <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>
%F A067151 Conjecture: a(n) ~ c * n^4. Is c = 1/64 ? - _Bill McEachen_, Mar 03 2024
%e A067151 a(6)=6 because the 6 regions around the center are quadrilaterals.
%Y A067151 Cf. A007678, A067163, A064869, A067152, A067153, A067154, A067155, A067156, A067157, A067158, A067159.
%K A067151 nonn
%O A067151 4,3
%A A067151 _Sascha Kurz_, Jan 06 2002
%E A067151 Title clarified, a(47) and above by _Scott R. Shannon_, Dec 04 2021