

A006600


Total number of triangles visible in regular ngon with all diagonals drawn.
(Formerly M4513)


15



1, 8, 35, 110, 287, 632, 1302, 2400, 4257, 6956, 11297, 17234, 25935, 37424, 53516, 73404, 101745, 136200, 181279, 236258, 306383, 389264, 495650, 620048, 772785, 951384, 1167453, 1410350, 1716191, 2058848, 2463384, 2924000, 3462305, 4067028, 4776219, 5568786, 6479551
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

3,2


COMMENTS

Place n equallyspaced points on a circle, join them in all possible ways; how many triangles can be seen?


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n=3..1000
Sascha Kurz, mgons in regular ngons
B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135156.
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135156 (1998).
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv version, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Mathematica programs for these sequences
D. Radcliffe, Counting triangles in a regular polygon
T. Sillke, Number of triangles for convex ngon
S. E. Sommars and T. Sommars, Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, J. Integer Sequences, 1 (1998), #98.1.5.
Sequences formed by drawing all diagonals in regular polygon


FORMULA

a(2n1) = A005732(2n1) for n > 1; a(2n) = A005732(2n)  A260417(n) for n > 1.  Jonathan Sondow, Jul 25 2015


EXAMPLE

a(4) = 8 because in a quadrilateral the diagonals cross to make four triangles, which pair up to make four more.


MATHEMATICA

del[m_, n_]:=If[Mod[n, m]==0, 1, 0]; Tri[n_]:=n(n1)(n2)(n^3+18n^243n+60)/720  del[2, n](n2)(n7)n/8  del[4, n](3n/4)  del[6, n](18n106)n/3 + del[12, n]*33n + del[18, n]*36n + del[24, n]*24n  del[30, n]*96n  del[42, n]*72n  del[60, n]*264n  del[84, n]*96n  del[90, n]*48n  del[120, n]*96n  del[210, n]*48n; Table[Tri[n], {n, 3, 1000}] (* T. D. Noe, Dec 21 2006 *)


CROSSREFS

Often confused with A005732.
Cf. A203016, A260417.
Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.
Sequence in context: A212674 A189592 A244882 * A161456 A005732 A162211
Adjacent sequences: A006597 A006598 A006599 * A006601 A006602 A006603


KEYWORD

nonn,easy,nice,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

a(3)a(8) computed by Victor Meally (personal communication to N. J. A. Sloane, circa 1975); later terms and recurrence from S. Sommars and T. Sommars.


STATUS

approved



