|
| |
|
|
A006600
|
|
Total number of triangles visible in regular n-gon with all diagonals drawn.
(Formerly M4513)
|
|
16
|
|
|
|
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 equally-spaced 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, m-gons in regular n-gons
Victor Meally, Letter to N. J. A. Sloane, no date.
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.
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, 135-156 (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
M. Rubinstein, Drawings for n=4,5,6,...
T. Sillke, Number of triangles for convex n-gon
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(2n-1) = A005732(2n-1) 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(n-1)(n-2)(n^3+18n^2-43n+60)/720 - del[2, n](n-2)(n-7)n/8 - del[4, n](3n/4) - del[6, n](18n-106)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: A189592 A285737 A244882 * A161456 A005732 A162211
Adjacent sequences: A006597 A006598 A006599 * A006601 A006602 A006603
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
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
|
| |
|
|
