A335102 - OEIS

%I #46 Jul 19 2024 11:34:10

%S 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 112,0,0,0,1,1365,1168,208,0,0,0,0,1,2380,1512,216,54,54,0,0,0,1,3876,

%U 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 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 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 Scott R. Shannon, <a href="/A335102/a335102.png">Image of the vertices for n=5</a>.

%H Scott R. Shannon, <a href="/A335102/a335102_1.png">Image of the vertices for n=6</a>.

%H Scott R. Shannon, <a href="/A335102/a335102_2.png">Image of the vertices for n=8</a>.

%H Scott R. Shannon, <a href="/A335102/a335102_3.png">Image of the vertices for n=12</a>.

%H Scott R. Shannon, <a href="/A335102/a335102_4.png">Image of the vertices for n=13</a>.

%H Scott R. Shannon, <a href="/A335102/a335102_5.png">Image of the vertices for n=16</a>.

%H Scott R. Shannon, <a href="/A335102/a335102_6.png">Image of the vertices for n=18</a>.

%H Scott R. Shannon, <a href="/A335102/a335102_7.png">Image of the vertices for n=20</a>.

%H Scott R. Shannon, <a href="/A335102/a335102_8.png">Image of the vertices for n=24</a>.

%H Scott R. Shannon, <a href="/A335102/a335102_9.png">Image of the vertices for n=30</a>.

%H <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>

%F 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 Table begins:

%e       0;

%e       0;

%e       0;

%e       1;

%e       5;

%e      12,    1;

%e      35;

%e      40,    8,   1;

%e     126;

%e     140,   20,   0,   1;

%e     330;

%e     228,   60,  12,   0,   1;

%e     715;

%e     644,  112,   0,   0,   0,  1;

%e    1365;

%e    1168,  208,   0,   0,   0,  0, 1;

%e    2380;

%e    1512,  216,  54,  54,   0,  0, 0, 1;

%e    3876;

%e    3360,  480,   0,   0,   0,  0, 0, 0, 1;

%e    5985;

%e    5280,  660,   0,   0,   0,  0, 0, 0, 0, 1;

%e    8855;

%e    6144,  864, 264,  24,   0,  0, 0, 0, 0, 0, 1;

%e   12650;

%e   11284, 1196,   0,   0,   0,  0, 0, 0, 0, 0, 0, 1;

%e   17550;

%e   15680, 1568,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 1;

%e   23751;

%e   13800, 2250, 420, 180, 120, 30, 0, 0, 0, 0, 0, 0, 0, 1;

%e   31465;

%e   28448, 2464,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

%e   40920;

%e   37264, 2992,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

%e   52360;

%Y Columns give A292104, A101363 (2n-gon), A101364, A101365.

%Y Row sums give A006561.

%Y Cf. A007569, A007678, A053126, A292105, A333275.

%K nonn,tabf

%O 1,5

%A _Scott R. Shannon_ and _N. J. A. Sloane_, May 23 2020