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A292105 Irregular triangle read by rows: T(n,k) = the number of interior points that are the intersections of exactly k chords in the configuration A006561(n) (n >= 1, k >= 1).

%I A292105 #40 Jun 08 2025 13:48:03
%S A292105 0,0,0,0,1,0,5,0,12,1,0,35,0,40,8,1,0,126,0,140,20,0,1,0,330,0,228,60,
%T A292105 12,0,1,0,715,0,644,112,0,0,0,1,0,1365,0,1168,208,0,0,0,0,1,0,2380,0,
%U A292105 1512,216,54,54,0,0,0,1,0,3876,0,3360,480,0,0,0,0,0,0,1,0,5985
%N A292105 Irregular triangle read by rows: T(n,k) = the number of interior points that are the intersections of exactly k chords in the configuration A006561(n) (n >= 1, k >= 1).
%H A292105 Seiichi Manyama, <a href="/A292105/b292105.txt">Rows n = 1..250, flattened</a>
%H A292105 B. Poonen and M. Rubinstein, <a href="http://arXiv.org/abs/math.MG/9508209">The number of intersection points made by the diagonals of a regular polygon</a>, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.
%H A292105 B. Poonen and M. Rubinstein, <a href="http://dx.doi.org/10.1137/S0895480195281246">Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156 (1998). [Copy on SIAM web site]
%H A292105 B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.pdf">The number of intersection points made by the diagonals of a regular polygon</a>, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998). [Copy on B. Poonen's web site]
%H A292105 B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.m">Mathematica programs for A006561 and related sequences</a>
%H A292105 N. J. A. Sloane, <a href="/A331450/a331450.jpg">Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19.</a> [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
%H A292105 N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, <a href="https://vimeo.com/237029685">Part I</a>, <a href="https://vimeo.com/237030304">Part 2</a>, <a href="https://oeis.org/A290447/a290447_slides.pdf">Slides.</a> (Mentions this sequence)
%H A292105 Scott R. Shannon, <a href="/A292105/a292105.txt">Table for n=1..100</a>.
%H A292105 Scott R. Shannon, <a href="/A292105/a292105_6.png">Image of 8-gon</a>.
%H A292105 Scott R. Shannon, <a href="/A292105/a292105_7.png">Image of 9-gon</a>.
%H A292105 Scott R. Shannon, <a href="/A292105/a292105_8.png">Image of 12-gon</a>.
%e A292105 Triangle begins:
%e A292105   0;
%e A292105   0;
%e A292105   0;
%e A292105   0,   1;
%e A292105   0,   5;
%e A292105   0,  12,  1;
%e A292105   0,  35;
%e A292105   0,  40,  8,  1;
%e A292105   0, 126;
%e A292105   0, 140, 20,  0, 1;
%e A292105   0, 330;
%e A292105   0, 228, 60, 12, 0, 1;
%e A292105 See the attached text file for the first 100 rows.
%Y A292105 Columns give A292104, A101363 (2n-gon), A101364, A101365.
%Y A292105 Row sums give A006561.
%Y A292105 Cf. A335102.
%K A292105 nonn,tabf
%O A292105 1,7
%A A292105 _N. J. A. Sloane_, Sep 14 2017
%E A292105 a(27) and beyond by _Scott R. Shannon_, May 15 2022