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.

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, 4, 51, 42, 400, 152, 154, 1, 1, 4, 61, 42, 680, 152, 802, 220, 1, 1, 4, 71, 42, 1030, 152, 1792, 590, 375, 1, 1, 4, 81, 42, 1450, 152, 2962, 690, 2091, 444, 1

Offset: 3

Author

Scott R. Shannon and N. J. A. Sloane, Dec 06 2021

Keywords

• nonn
• tabl

Comments

We use "cell" in the sense of planar graph theory, meaning a "region" or two-dimensional face.

We start at generation 0 with a regular n-gon with a single cell.

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.

For the first few generations of mitosis of a triangle, square, pentagon, and hexagon, see the sketch in one of the links below.

The process of going from generation 0 to generation 1 was analyzed by Poonen and Rubinstein (1998) - see A007678 and A331450.

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.

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.

For example, T(11,k) = 220*k^2 + 1452*k - 1693 for k >= 2. See the Formulas section below for further examples.

Note that if n is odd, all generations of mitosis of a regular n-gon contain a (smaller) regular n-gon at their center.

Conjecture 2: Apart from the central n-gon when n is odd, any cell will eventually split into a mixture of triangles and pentagons.

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.

Examples

			The table begins:
.
      |               Number of polygons after k generations
  n\k | 0,    1,     2,     3,      4,      5,      6,      7,      8,      9, ...
----------------------------------------------------------------------------------
   3  | 1,    1,     1,     1,      1,      1,      1,      1,      1,      1, ...
   4  | 1,    4,     4,     4,      4,      4,      4,      4,      4,      4, ...
   5  | 1,   11,    21,    31,     41,     51,     61,     71,     81,     91, ...
   6  | 1,   24,    42,    42,     42,     42,     42,     42,     42,     42, ...
   7  | 1,   50,   190,   400,    680,   1030,   1450,   1940,   2500,   3130, ...
   8  | 1,   80,   152,   152,    152,    152,    152,    152,    152,    152, ...
   9  | 1,  154,   802,  1792,   2962,   4312,   5842,   7552,   9442,  11512, ...
  10  | 1,  220,   590,   690,    790,    890,    990,   1090,   1190,   1290, ...
  11  | 1,  375,  2091,  4643,   7635,  11067,  14939,  19251,  24003,  29195, ...
  12  | 1,  444,   948,   948,    948,    948,    948,    948,    948,    948, ...
  13  | 1,  781,  5461, 14119,  24727,  37285,  51793,  68251,  86659, 107017, ...
  14  | 1,  952,  3066,  4046,   5026,   6006,   6986,   7966,   8946,   9926, ...
  15  | 1, 1456,  9361, 22756,  40186,  61066,  85396, 113176, 144406, 179086, ...
  16  | 1, 1696,  6096,  8240,   9520,  10800,  12080,  13360,  14640,  15920, ...
  17  | 1, 2500, 18225, 49131,  90883, 143175, 206007, 279379, 363291, 457743, ...
  18  | 1, 2466,  7344, 10872,  14166,  16866,  19566,  22266,  24966,  27666, ...
  19  | 1, 4029, 29356, 77616, 140316, 217456, 309036, 415056, 535516, 670416, ...
  20  | 1, 4500, 19580, 31620,  39820,  48020,  56220,  64420,  72620,  80820, ...
  21  | 1, 6175, 40720, 97336, 168022, 252778, 351604, 464500, 591466, 732502, ...
  22  | 1, 6820, 31042, 52030,  65890,  79750,  93610, 107470, 121330, 135190, ...
.
		

Links

• B. Poonen and M. Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11, no. 1 (1998), pp. 135-156; DOI:10.1137/S0895480195281246. [Author's copy]. The latest arXiv version arXiv:math/9508209 has corrected some typos in the published version.
• Scott R. Shannon, Extended Table of A350000 for 5 <= n <= 46, Dec 22, 2021 [This shows the initial terms of the rows in human order (not by antidiagonals)]
• Scott R. Shannon, Illustration for T(9,1)
• Scott R. Shannon, T(9,2)
• Scott R. Shannon, T(9,3)
• Scott R. Shannon, T(9,4)
• Scott R. Shannon, T(10,1)
• Scott R. Shannon, T(10,2)
• Scott R. Shannon, T(10,3)
• Scott R. Shannon, T(11,1)
• Scott R. Shannon, T(11,2)
• Scott R. Shannon, T(11,3)
• Scott R. Shannon, T(14,1)
• Scott R. Shannon, T(14,2)
• Scott R. Shannon, T(14,3)
• Scott R. Shannon, T(17,3)
• Scott R. Shannon, T(29,1)
• Scott R. Shannon, Close-up of the 11-gon in T(29,1)
• Scott R. Shannon, Close-up of a 9-gon in T(29,2). 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.
• Scott R. Shannon, Zoomed-in view of T(51,2). This shows the complicated structure formed after just 2 generations, typical of larger values of n.
• N. J. A. Sloane, Rough sketch of first few generations of mitosis of a triangle, square, pentagon, and hexagon. 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.

Crossrefs

Cf. A007678 (column 1), A349807 (column 2), A017281 (row 5), A349808 (row 7); also A350501, A350502.

Cf. also A331450, A349967, A349968.

Formula

Formulas for the initial rows: (These are easy to prove.)

To avoid double subscripts, we use a(k) for T(n,k) when we are looking at row n.

n=3: a(k) = 1, for k >= 0.

n=4: a(0) = 1, a(k) = 4 for k >= 1.

n=5: a(k) = 10k+1, k >= 0. See A017281.

n=6: a(0) = 1, a(1) = 24, a(k) = 42 for k >= 2.

n=7: a(0) = 1, a(k) = 35*k^2+35*k-20 for k >= 1. See A349808.

n=8: a(0) = 1, a(1) = 80, a(k) = 152 for k >= 2.

n=9: a(0) = 1, a(1) = 154, a(k) = 90*k^2+540*k-638 for k >= 2.

n=10: a(0) = 1, a(1) = 220, a(k) = 100*k+390 for k >= 2.

n=11: a(0) = 1, a(1) = 375, a(k) = 220*k^2 + 1452*k - 1693 for k >= 2.

n=12: a(0) = 1, a(1) = 444, a(k) = 948 for k >= 2.

n=13: a(0) = 1, a(1) = 781, a(k) = 975*k^2 + 3783*k - 6005 for k >= 2.

n=14: a(0) = 1, a(k) = 980*k + 1106 for k >= 1.

n=15: a(k) = 1725*k^2+5355*k-8834 for k >= 3.

n=16: a(k) = 1280*k + 4400 for k >= 3.

n=18: a(k) = 2700*k + 3366 for k >= 4.

Also T(n,1) = A007678(n).