login
Number of triple-crossings of diagonals in the regular 2n-gon.
4

%I #14 Jul 25 2015 17:11:26

%S 0,1,12,30,128,147,264,1056,600,825,2380,1482,1932,9635,3024,3672,

%T 8484,5301,6300,19474,8580,9867,20744,12900,14664,30141,18564,20706,

%U 62200,25575,28320,54956,34272,37485,62868,44622,48564,86359,57000,61500,117068,71337

%N Number of triple-crossings of diagonals in the regular 2n-gon.

%C Same as (total number of triangles visible in convex 2n-gon with all diagonals drawn in general position) - (total number of triangles visible in regular 2n-gon with all diagonals drawn).

%C Number of triple-crossings of diagonals in the regular 2n+1-gon is 0.

%C See Sillke 1998 (where a(n) is called "T(2n)") for explanations and extensive annotated references.

%C See A005732 and A006600 for more comments, references, links, formulas, examples, programs, and lists from which to compute a(n) = A005732(2n) - A006600(2n) up to n = 500.

%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/triangle_counting">Number of triangles for a convex n-gon</a>, 1998.

%F a(n) = A005732(2n) - A006600(2n).

%e With only 2 diagonals in a 4-gon, there can be no triple-crossings, so a(2) = 0.

%Y Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

%K nonn

%O 2,3

%A _Jonathan Sondow_, Jul 25 2015