A260417 -id:A260417 - cp's OEIS frontend

A006600 Total number of triangles visible in regular n-gon with all diagonals drawn.

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

Offset: 3

N. J. A. Sloane

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

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

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

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
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

Often confused with A005732.
Cf. A203016, A260417.
Row sums of A363174.
Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

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 *)

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

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.

A005732 a(n) = binomial(n+3,6) + binomial(n+1,5) + binomial(n,5).

Original entry on oeis.org

1, 8, 35, 111, 287, 644, 1302, 2430, 4257, 7084, 11297, 17381, 25935, 37688, 53516, 74460, 101745, 136800, 181279, 237083, 306383, 391644, 495650, 621530, 772785, 953316, 1167453, 1419985, 1716191, 2061872, 2463384, 2927672, 3462305, 4075512, 4776219, 5574087, 6479551

Offset: 3

N. J. A. Sloane

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

Equals binomial transform of [1, 7, 20, 29, 22, 8, 1, 0, 0, 0, ...]. - Gary W. Adamson, Jun 13 2008

C. L. Liu, Introduction to Combinatorial Analysis. McGraw-Hill, NY, 1968, p. 20.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 3..1000
Claudi Alsina and Roger B. Nelson, A Panoply of Polygons, Dolciani Math. Expeditions, AMS/MAA (2023) Vol. 58, see page 7.
R. J. Cormier and R. B. Eggleton, Counting by correspondence, Math. Mag., 49 (1976), 181-186.
R. K. Guy & M. E. Larsen, Correspondence, 1986-87
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
T. Sillke, Number of triangles for a convex n-gon
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Often confused with A006600.

Cf. A007318, A260417.

a005732 n = a005732_list !! (n-3)
a005732_list = 1 : 8 : f (drop 5 a007318_tabl) where
   f (us:pss@(vs::ws:)) = (us !! 5 + vs !! 5 + ws !! 6) : f pss
-- Reinhard Zumkeller, Mar 11 2014

[Binomial(n+3, 6) + Binomial(n+1, 5) +Binomial(n,5): n in [3..100]]; // Vincenzo Librandi, Apr 10 2011

Table[Binomial[n+3,6]+Binomial[n+1,5]+Binomial[n,5],{n,3,40}]  (* Harvey P. Dale, Apr 09 2011 *)

a(n)=binomial(n+3,6) + binomial(n+1,5) + binomial(n,5) \\ Charles R Greathouse IV, Feb 19 2017

G.f.: x^3*(-1-x+x^3) / (x-1)^7 . - Simon Plouffe in his 1992 dissertation

a(2n-1) = A006600(2n-1) for n > 1; a(2n) = A006600(2n) + A260417(n) for n > 1. - Jonathan Sondow, Jul 25 2015

Thanks to Joshua Zucker, Ted Alper and Joe Keane for clarifying the connection with A006600.

A363174 Array read by rows: T(n,k) is the number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from k distinct vertices, with n >= 3, 3 <= k <= 6.

Original entry on oeis.org

1, 0, 0, 0, 4, 4, 0, 0, 10, 20, 5, 0, 20, 60, 30, 0, 35, 140, 105, 7, 56, 280, 280, 16, 84, 504, 630, 84, 120, 840, 1260, 180, 165, 1320, 2310, 462, 220, 1980, 3960, 796, 286, 2860, 6435, 1716, 364, 4004, 10010, 2856, 455, 5460, 15015, 5005, 560, 7280, 21840, 7744

Offset: 3

Paolo Xausa, May 19 2023

See Sommars and Sommars (1998) for a complete analysis of the problem.

			Array begins:
  n\k|     3     4     5     6
  ---+---------------------------
   3 |     1,    0,    0,    0;
   4 |     4,    4,    0,    0;
   5 |    10,   20,    5,    0;
   6 |    20,   60,   30,    0;
   7 |    35,  140,  105,    7;
   8 |    56,  280,  280,   16;
   9 |    84,  504,  630,   84;
  10 |   120,  840, 1260,  180;
  ...

Paolo Xausa, Table of n, a(n) for n = 3..10002 (rows 3..2502 of array, flattened).
Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006.
Steven E. Sommars and Tim Sommars, The Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.5.
Paolo Xausa, Illustration of T(9,6).
Sequences formed by drawing all diagonals in regular polygon

Cf. A000579, A006561, A006600 (row sums), A260417.

Cf. A000292 (column k = 3), A033488 (column k = 4), A174002 (column k = 5), A363173 (column k = 6).

A363174list[rowmax_]:=Module[{d},d[m_,n_]:=Boole[Divisible[n,m]];Table[Binomial[n,k]If[4<=k<=5,k,1]-If[k==6&&EvenQ[n],((1/8n^2-9/8n+7/4)d[2,n]+3/4d[4,n]+(6n-106/3)d[6,n]-33d[12,n]-36d[18,n]-24d[24,n]+96d[30,n]+72d[42,n]+264d[60,n]+96d[84,n]+48d[90,n]+96d[120,n]+48d[210,n])n,0],{n,3,rowmax},{k,3,6}]];A363174list[20]

T(n,3) = binomial(n,3) = A000292(n-2).
T(n,4) = 4*binomial(n,4) = A033488(n-3).
T(n,5) = 5*binomial(n,5) = A174002(n-4), for n >= 4.
T(n,6) = binomial(n,6) = A000579(n) if n is odd, A000579(n) - A260417(n/2) if n is even.
Sum_{k=3..6} T(n,k) = A006600(n).

A363173 Number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from 6 distinct vertices.

Original entry on oeis.org

0, 0, 0, 0, 7, 16, 84, 180, 462, 796, 1716, 2856, 5005, 7744, 12376, 17508, 27132, 38160, 54264, 73788, 100947, 132216, 177100, 228748, 296010, 374808, 475020, 584140, 736281, 903168, 1107568, 1341232, 1623160, 1939308, 2324784, 2755380, 3262623, 3832080, 4496388

Offset: 3

Paolo Xausa, May 19 2023

nonn

Paolo Xausa, Table of n, a(n) for n = 3..10000
Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006.
Steven E. Sommars and Tim Sommars, The Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.5.
Paolo Xausa, Illustration of a(9).
Sequences formed by drawing all diagonals in regular polygon

Column k = 6 of A363174.

Cf. A000579, A006561, A260417.

A363173list[nmax_]:=Module[{d},d[m_,n_]:=Boole[Divisible[n,m]];Table[Binomial[n, 6]-If[EvenQ[n],((1/8n^2-9/8n+7/4)d[2,n]+3/4d[4,n]+(6n-106/3)d[6,n]-33d[12,n]-36d[18,n]-24d[24,n]+96d[30,n]+72d[42,n]+264d[60,n]+96d[84,n]+48d[90,n]+96d[120,n]+48d[210,n])n,0],{n,3,nmax}]];A363173list[50]

a(n) = binomial(n,6) = A000579(n) if n is odd, A000579(n) - A260417(n/2) if n is even.

cp's OEIS Frontend

A006600 Total number of triangles visible in regular n-gon with all diagonals drawn.

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A005732 a(n) = binomial(n+3,6) + binomial(n+1,5) + binomial(n,5).

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A363174 Array read by rows: T(n,k) is the number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from k distinct vertices, with n >= 3, 3 <= k <= 6.

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A363173 Number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from 6 distinct vertices.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula