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A062361 Number of triangular regions in regular n-gon with all diagonals drawn.

%I A062361 #32 Jun 08 2025 13:50:57
%S A062361 1,4,10,18,35,56,90,120,176,276,377,476,585,848,1054,1404,1653,2200,
%T A062361 2268,2992,3749,4416,5000,6292,6777,8316,9222,11670,11501,14368,15840,
%U A062361 18598,19705,24444,25012,28842,30966,36000,39278,45318,46999,53900
%N A062361 Number of triangular regions in regular n-gon with all diagonals drawn.
%C A062361 Also the number of 3-cycles and maximum cliques in the n-polygon diagonal intersection graph. - _Eric W. Weisstein_, Mar 08-09 2018
%H A062361 Andrew Howroyd, <a href="/A062361/b062361.txt">Table of n, a(n) for n = 3..100</a>
%H A062361 B. Poonen and M. Rubinstein, <a href="https://doi.org/10.1137/S0895480195281246">Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.
%H A062361 B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.pdf">The number of intersection points made by the diagonals of a regular polygon</a>, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
%H A062361 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 A062361 S. E. Sommars and T. Sommars, <a href="http://www.cs.uwaterloo.ca/journals/JIS/sommars/newtriangle.html">Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon</a>, J. Integer Sequences, 1 (1998), #98.1.5.
%H A062361 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H A062361 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a>
%H A062361 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonDiagonalIntersectionGraph.html">Polygon Diagonal Intersection Graph</a>
%H A062361 <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>
%F A062361 a(n) = n * A067162(n).
%e A062361 a(4) = 4 because in a quadrilateral the diagonals cross to make four triangles.
%Y A062361 Cf. A006600, A007678.
%Y A062361 Cf. A300552 (4-cycles), A300553 (5-cycles), A300554 (6-cycles).
%K A062361 easy,nonn
%O A062361 3,2
%A A062361 _Sascha Kurz_, Jul 07 2001