小学生的难题

northwolves

郁闷。昨天咋没搜到呢？
A006600  Triangles in regular n-gon.  



A006600  Triangles in regular n-gon.
(Formerly M4513)   +0
12  

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; listen)  

OFFSET  3,2
  
COMMENT  Place n equally-spaced points on a circle, join them in all possible ways; how many triangles can be seen?

LINKS  T. D. Noe, Table of n, a(n) for n=3..1000

Sascha Kurz, m-gons in regular n-gons

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

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

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 (noe(AT)sspectra.com), Dec 21 2006

CROSSREFS  Often confused with A005732.

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

Adjacent sequences: A006597 A006598 A006599 this_sequence A006601 A006602 A006603

Sequence in context: A136016 A100907 A058102 this_sequence A005732 A040977 A036598

KEYWORD  nonn,easy,nice

AUTHOR  njas

EXTENSIONS  a(3)...a(8) computed by Victor Meally (personal communication); later terms and recurrence from S. Sommars and T. Sommars.

mathe

比如n=5,那么i)=$C_5^3=10$;ii)=$4*C_5^4=20$;iii)=$5*C_4^4=5$,iv)=$C_5^6=0$,而且没有三点共线,所以总共10+20=5=35.
对于n=6,那么i)=$C_6^3=20$;ii)=$4*C_6^4=60$;iii)=$6*C_5^4=30$,iv)=$C_6^6=1$,三点共线1组,所以总共20+60+30+1-1=110

mathe

原帖由 northwolves 于 2009-1-9 15:19 发表
郁闷。昨天咋没搜到呢？
A006600  Triangles in regular n-gon.  



A006600  Triangles in regular n-gon.
(Formerly M4513)   +0
12  

1, 8, 35, 110, 287, 632, 1302, 2400, 4257, 6956, 11297,  ...

呵呵,没有前面几个的准确数据还是比较难搜到的.
我也挺郁闷,刚刚导出部分公式,你正好搜到了

无心人

6的不存在内点三角形吧??

mathe

原帖由 无心人 于 2009-1-9 15:25 发表
6的不存在内点三角形吧??

正好3线共点.所以我上面的计算过程为先加1再减1.

无心人

另外
对mathe的三点共线一直有疑问
应该是三线共点吧

mathe

原帖由 无心人 于 2009-1-9 15:28 发表
另外
对mathe的三点共线一直有疑问
应该是三线共点吧

对,全部是三线共点,没有三点共线

northwolves

这个网页解释得比较不错,图文并茂：

The Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon

无心人

你还继续分析么?
被提前给出公式的题目
我想类似你正在解题的时候
有人捉住你头发拉起来你一样
估计没兴趣了吧

mathe

原帖由 无心人 于 2009-1-9 15:36 发表


你还继续分析么?
被提前给出公式的题目
我想类似你正在解题的时候
有人捉住你头发拉起来你一样
估计没兴趣了吧

当然到此为止了
其实通常情况还好.不过这次可是将所有需要的东西都准备好了