A363174 - OEIS

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.

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

3,5

COMMENTS

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

LINKS

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

FORMULA

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).

EXAMPLE

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;

...

MATHEMATICA

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]

CROSSREFS

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

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

Sequence in context: A190113 A165727 A284609 * A290448 A282593 A276339

Adjacent sequences: A363171 A363172 A363173 * A363175 A363176 A363177

KEYWORD

nonn,tabl

AUTHOR

Paolo Xausa, May 19 2023

STATUS

approved