cp's OEIS Frontend

A335102 Irregular triangle read by rows: consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.

%I A335102 #46 Jul 19 2024 11:34:10
%S A335102 0,0,0,1,5,12,1,35,40,8,1,126,140,20,0,1,330,228,60,12,0,1,715,644,
%T A335102 112,0,0,0,1,1365,1168,208,0,0,0,0,1,2380,1512,216,54,54,0,0,0,1,3876,
%U A335102 3360,480,0,0,0,0,0,0,1,5985,5280,660,0,0,0,0,0,0,0,1,8855,6144,864,264,24,0,0,0,0,0,0,12,12650
%N A335102 Irregular triangle read by rows:  consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.
%H A335102 B. Poonen and M. Rubinstein, <a href="https://arxiv.org/abs/math/9508209">The number of intersection points made by the diagonals of a regular polygon</a>, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
%H A335102 Scott R. Shannon, <a href="/A335102/a335102.png">Image of the vertices for n=5</a>.
%H A335102 Scott R. Shannon, <a href="/A335102/a335102_1.png">Image of the vertices for n=6</a>.
%H A335102 Scott R. Shannon, <a href="/A335102/a335102_2.png">Image of the vertices for n=8</a>.
%H A335102 Scott R. Shannon, <a href="/A335102/a335102_3.png">Image of the vertices for n=12</a>.
%H A335102 Scott R. Shannon, <a href="/A335102/a335102_4.png">Image of the vertices for n=13</a>.
%H A335102 Scott R. Shannon, <a href="/A335102/a335102_5.png">Image of the vertices for n=16</a>.
%H A335102 Scott R. Shannon, <a href="/A335102/a335102_6.png">Image of the vertices for n=18</a>.
%H A335102 Scott R. Shannon, <a href="/A335102/a335102_7.png">Image of the vertices for n=20</a>.
%H A335102 Scott R. Shannon, <a href="/A335102/a335102_8.png">Image of the vertices for n=24</a>.
%H A335102 Scott R. Shannon, <a href="/A335102/a335102_9.png">Image of the vertices for n=30</a>.
%H A335102 <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>
%F A335102 If n = 2t+1 is odd then the n-th row has a single term, T(2t+1, 2t+4) = binomial(2t+1,4) (these values are given in A053126).
%e A335102 Table begins:
%e A335102       0;
%e A335102       0;
%e A335102       0;
%e A335102       1;
%e A335102       5;
%e A335102      12,    1;
%e A335102      35;
%e A335102      40,    8,   1;
%e A335102     126;
%e A335102     140,   20,   0,   1;
%e A335102     330;
%e A335102     228,   60,  12,   0,   1;
%e A335102     715;
%e A335102     644,  112,   0,   0,   0,  1;
%e A335102    1365;
%e A335102    1168,  208,   0,   0,   0,  0, 1;
%e A335102    2380;
%e A335102    1512,  216,  54,  54,   0,  0, 0, 1;
%e A335102    3876;
%e A335102    3360,  480,   0,   0,   0,  0, 0, 0, 1;
%e A335102    5985;
%e A335102    5280,  660,   0,   0,   0,  0, 0, 0, 0, 1;
%e A335102    8855;
%e A335102    6144,  864, 264,  24,   0,  0, 0, 0, 0, 0, 1;
%e A335102   12650;
%e A335102   11284, 1196,   0,   0,   0,  0, 0, 0, 0, 0, 0, 1;
%e A335102   17550;
%e A335102   15680, 1568,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 1;
%e A335102   23751;
%e A335102   13800, 2250, 420, 180, 120, 30, 0, 0, 0, 0, 0, 0, 0, 1;
%e A335102   31465;
%e A335102   28448, 2464,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A335102   40920;
%e A335102   37264, 2992,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A335102   52360;
%Y A335102 Columns give A292104, A101363 (2n-gon), A101364, A101365.
%Y A335102 Row sums give A006561.
%Y A335102 Cf. A007569, A007678, A053126, A292105, A333275.
%K A335102 nonn,tabf
%O A335102 1,5
%A A335102 _Scott R. Shannon_ and _N. J. A. Sloane_, May 23 2020