This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350000 #46 Mar 02 2022 10:05:20 %S A350000 1,1,1,1,4,1,1,4,11,1,1,4,21,24,1,1,4,31,42,50,1,1,4,41,42,190,80,1,1, %T A350000 4,51,42,400,152,154,1,1,4,61,42,680,152,802,220,1,1,4,71,42,1030,152, %U A350000 1792,590,375,1,1,4,81,42,1450,152,2962,690,2091,444,1 %N A350000 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of cells in a regular n-gon after k generations of mitosis. %C A350000 We use "cell" in the sense of planar graph theory, meaning a "region" or two-dimensional face. %C A350000 We start at generation 0 with a regular n-gon with a single cell. %C A350000 At each stage the mitosis process splits each cell into smaller cells by drawing chords between every pair of points on the boundary of that cell. %C A350000 For the first few generations of mitosis of a triangle, square, pentagon, and hexagon, see the sketch in one of the links below. %C A350000 The process of going from generation 0 to generation 1 was analyzed by Poonen and Rubinstein (1998) - see A007678 and A331450. %C A350000 It is worth enlarging the illustrations in order to see the detailed structure and the cell counts in the upper left corner. The illustrations for the mitosis of a 7-gon can be seen in A349808 and are not repeated here. %C A350000 Conjecture 1: For a fixed value of n, there are integers r and s, which are small compared to n, such that T(n,k) is a polynomial in k of degree r for all k >= s. %C A350000 For example, T(11,k) = 220*k^2 + 1452*k - 1693 for k >= 2. See the Formulas section below for further examples. %C A350000 Note that if n is odd, all generations of mitosis of a regular n-gon contain a (smaller) regular n-gon at their center. %C A350000 Conjecture 2: Apart from the central n-gon when n is odd, any cell will eventually split into a mixture of triangles and pentagons. %C A350000 If we think of triangles and pentagons are harmless cells, and all other cells as dangerous, the conjecture states that (with the exception of the central odd cells), all cells eventually become harmless. %H A350000 B. Poonen and M. Rubinstein, <a href="https://doi.org/10.1137/S0895480195281246">The Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics 11, no. 1 (1998), pp. 135-156; DOI:10.1137/S0895480195281246. [<a href="http://math.mit.edu/~poonen/papers/ngon.pdf">Author's copy</a>]. The latest arXiv version <a href="http://arxiv.org/abs/math/9508209">arXiv:math/9508209</a> has corrected some typos in the published version. %H A350000 Scott R. Shannon, <a href="/A350000/a350000.txt">Extended Table of A350000 for 5 <= n <= 46</a>, Dec 22, 2021 [This shows the initial terms of the rows in human order (not by antidiagonals)] %H A350000 Scott R. Shannon, <a href="/A350000/a350000.gif">Illustration for T(9,1)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_1.gif">T(9,2)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_2.gif">T(9,3)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_3.gif">T(9,4)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_4.gif">T(10,1)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_5.gif">T(10,2)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_6.gif">T(10,3)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_7.gif">T(11,1)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_8.gif">T(11,2)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_9.gif">T(11,3)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_10.gif">T(14,1)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_11.gif">T(14,2)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_12.gif">T(14,3)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_13.gif">T(17,3)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_14.gif">T(29,1)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000.jpg">Close-up of the 11-gon in T(29,1)</a> %H A350000 Scott R. Shannon, <a href="/A350000/a350000_1.jpg">Close-up of a 9-gon in T(29,2)</a>. This shows the mitosis of the 11-gon from generation 1 in the previous image -- it has formed one 9-gon, five 7-gons, twelve 6-gons and numerous other 5, 4, and 3-gons. %H A350000 Scott R. Shannon, <a href="/A350000/a350000.png">Zoomed-in view of T(51,2)</a>. This shows the complicated structure formed after just 2 generations, typical of larger values of n. %H A350000 N. J. A. Sloane, <a href="/A350000/a350000.pdf">Rough sketch of first few generations of mitosis of a triangle, square, pentagon, and hexagon.</a> The central pentagonal cell of the pentagon splits into 10 triangles and a pentagon at every generation, with the cells getting smaller and smaller. The third splitting is drawn in red ink. The second splitting of the hexagon is also drawn in red ink, but then all the cells are triangles, and no further mitosis takes place. %F A350000 Formulas for the initial rows: (These are easy to prove.) %F A350000 To avoid double subscripts, we use a(k) for T(n,k) when we are looking at row n. %F A350000 n=3: a(k) = 1, for k >= 0. %F A350000 n=4: a(0) = 1, a(k) = 4 for k >= 1. %F A350000 n=5: a(k) = 10k+1, k >= 0. See A017281. %F A350000 n=6: a(0) = 1, a(1) = 24, a(k) = 42 for k >= 2. %F A350000 n=7: a(0) = 1, a(k) = 35*k^2+35*k-20 for k >= 1. See A349808. %F A350000 n=8: a(0) = 1, a(1) = 80, a(k) = 152 for k >= 2. %F A350000 n=9: a(0) = 1, a(1) = 154, a(k) = 90*k^2+540*k-638 for k >= 2. %F A350000 n=10: a(0) = 1, a(1) = 220, a(k) = 100*k+390 for k >= 2. %F A350000 n=11: a(0) = 1, a(1) = 375, a(k) = 220*k^2 + 1452*k - 1693 for k >= 2. %F A350000 n=12: a(0) = 1, a(1) = 444, a(k) = 948 for k >= 2. %F A350000 n=13: a(0) = 1, a(1) = 781, a(k) = 975*k^2 + 3783*k - 6005 for k >= 2. %F A350000 n=14: a(0) = 1, a(k) = 980*k + 1106 for k >= 1. %F A350000 n=15: a(k) = 1725*k^2+5355*k-8834 for k >= 3. %F A350000 n=16: a(k) = 1280*k + 4400 for k >= 3. %F A350000 n=18: a(k) = 2700*k + 3366 for k >= 4. %F A350000 Also T(n,1) = A007678(n). %e A350000 The table begins: %e A350000 . %e A350000 | Number of polygons after k generations %e A350000 n\k | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ... %e A350000 ---------------------------------------------------------------------------------- %e A350000 3 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... %e A350000 4 | 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, ... %e A350000 5 | 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, ... %e A350000 6 | 1, 24, 42, 42, 42, 42, 42, 42, 42, 42, ... %e A350000 7 | 1, 50, 190, 400, 680, 1030, 1450, 1940, 2500, 3130, ... %e A350000 8 | 1, 80, 152, 152, 152, 152, 152, 152, 152, 152, ... %e A350000 9 | 1, 154, 802, 1792, 2962, 4312, 5842, 7552, 9442, 11512, ... %e A350000 10 | 1, 220, 590, 690, 790, 890, 990, 1090, 1190, 1290, ... %e A350000 11 | 1, 375, 2091, 4643, 7635, 11067, 14939, 19251, 24003, 29195, ... %e A350000 12 | 1, 444, 948, 948, 948, 948, 948, 948, 948, 948, ... %e A350000 13 | 1, 781, 5461, 14119, 24727, 37285, 51793, 68251, 86659, 107017, ... %e A350000 14 | 1, 952, 3066, 4046, 5026, 6006, 6986, 7966, 8946, 9926, ... %e A350000 15 | 1, 1456, 9361, 22756, 40186, 61066, 85396, 113176, 144406, 179086, ... %e A350000 16 | 1, 1696, 6096, 8240, 9520, 10800, 12080, 13360, 14640, 15920, ... %e A350000 17 | 1, 2500, 18225, 49131, 90883, 143175, 206007, 279379, 363291, 457743, ... %e A350000 18 | 1, 2466, 7344, 10872, 14166, 16866, 19566, 22266, 24966, 27666, ... %e A350000 19 | 1, 4029, 29356, 77616, 140316, 217456, 309036, 415056, 535516, 670416, ... %e A350000 20 | 1, 4500, 19580, 31620, 39820, 48020, 56220, 64420, 72620, 80820, ... %e A350000 21 | 1, 6175, 40720, 97336, 168022, 252778, 351604, 464500, 591466, 732502, ... %e A350000 22 | 1, 6820, 31042, 52030, 65890, 79750, 93610, 107470, 121330, 135190, ... %e A350000 . %Y A350000 Cf. A007678 (column 1), A349807 (column 2), A017281 (row 5), A349808 (row 7); also A350501, A350502. %Y A350000 Cf. also A331450, A349967, A349968. %K A350000 nonn,tabl %O A350000 3,5 %A A350000 _Scott R. Shannon_ and _N. J. A. Sloane_, Dec 06 2021