A356799 Table read by antidiagonals: T(n,k) (n >= 2, k >= 1) is the number of regions formed in a regular 2n-gon by straight line segments when connecting the k+1 points that divide each side into k equal parts to the equivalent point on the side diagonally opposite.

1, 4, 13, 9, 24, 25, 16, 55, 48, 41, 25, 66, 105, 70, 61, 36, 121, 144, 171, 108, 85, 49, 126, 233, 220, 253, 140, 113, 64, 211, 288, 381, 312, 351, 192, 145, 81, 204, 409, 450, 565, 448, 465, 234, 181, 100, 325, 480, 671, 636, 785, 608, 595, 300, 221, 121, 300, 633, 760, 997, 924, 1041, 738, 741, 352, 265

Offset: 2

Author

Scott R. Shannon, Aug 28 2022

Keywords

• nonn
• tabl

Comments

Many rows and columns in the table appear to be given by a quadratic in even and odd values of k and n; see the Formula section. The exceptions are for rows with n mod 6 = 0 for even k, and for columns with even k, formulas for which are unknown.

Examples

			The table begins:
    1,   4,    9,   16,   25,   36,   49,    64,    81,   100,   121,   144, ...
   13,  24,   55,   66,  121,  126,  211,   204,   325,   300,   463,   414, ...
   25,  48,  105,  144,  233,  288,  409,   480,   633,   720,   905,  1008, ...
   41,  70,  171,  220,  381,  450,  671,   760,  1041,  1150,  1491,  1620, ...
   61, 108,  253,  312,  565,  636,  997,  1056,  1549,  1596,  2221,  2232, ...
   85, 140,  351,  448,  785,  924, 1387,  1568,  2157,  2380,  3095,  3360, ...
  113, 192,  465,  608, 1041, 1248, 1841,  2112,  2865,  3200,  4113,  4512, ...
  145, 234,  595,  738, 1333, 1512, 2359,  2556,  3673,  3870,  5275,  5454, ...
  181, 300,  741,  960, 1661, 1980, 2941,  3360,  4581,  5100,  6581,  7200, ...
  221, 352,  903, 1144, 2025, 2376, 3587,  4048,  5589,  6160,  8031,  8712, ...
  265, 432, 1081, 1344, 2425, 2784, 4297,  4704,  6697,  7152,  9625, 10080, ...
  313, 494, 1275, 1612, 2861, 3354, 5071,  5720,  7905,  8710, 11363, 12324, ...
  365, 588, 1485, 1904, 3333, 3948, 5909,  6720,  9213, 10220, 13245, 14448, ...
  421, 660, 1711, 2130, 3841, 4410, 6811,  7500, 10621, 11400, 15271, 16110, ...
  481, 768, 1953, 2496, 4385, 5184, 7777,  8832, 12129, 13440, 17441, 19008, ...
  545, 850, 2211, 2788, 4965, 5814, 8807,  9928, 13737, 15130, 19755, 21420, ...
  613, 972, 2485, 3096, 5581, 6444, 9901, 10944, 15445, 16668, 22213, 23544, ...
  .
  .
		

Links

• Scott R. Shannon, Table for n=2..35, k=1..50.
• Scott R. Shannon, Image for T(2,8) = 64.
• Scott R. Shannon, Image for T(3,6) = 126.
• Scott R. Shannon, Image for T(3,7) = 211.
• Scott R. Shannon, Image for T(4,8) = 480.
• Scott R. Shannon, Image for T(4,9) = 633.
• Scott R. Shannon, Image for T(6,12) = 2232.
• Scott R. Shannon, Image for T(6,13) = 3013.
• Scott R. Shannon, Image for T(10,10) = 5100.
• Scott R. Shannon, Image for T(10,11) = 6581.
• Scott R. Shannon, Image for T(18,1) = 613.
• Scott R. Shannon, Image for T(18,2) = 972.
• Scott R. Shannon, Image for T(18,3) = 2485.
• Scott R. Shannon, Image for T(18,4) = 3096.
• Scott R. Shannon, Image for T(18,5) = 5581.
• Scott R. Shannon, Image for T(18,6) = 6444.

Crossrefs

Cf. A000217, A265225, A000096, A000290, A005563, A001844, A001105, A051890, A249127, A356044.

Formula

T(2,k) = k^2.

Conjectured formula for the rows for odd values of k for n>=3:

T(n,k) = A000217(n-1)*k^2 + n^2*k + A000217(n-2) = (n^2 - n)*k^2/2 + n^2*k + (n^2 - 3n + 2)/2.

E.g., T(7,k) = A000217(6)*k^2 + 7^2*k + A000217(5) = 21k^2 + 49k + 15.

Conjectured formula for the rows for even values of k for n>=3:

For n mod 3 = 1 or n mod 3 = 2, T(n,k) = A000217(n-1)*k^2 + A265225(n-1)*k = (n^2 - n)*k^2/2 + (floor(n/2) + 1)*n*k.

E.g., T(10,k) = A000217(9)*k^2 + A265225(9)*k = 45k^2 + 60k.

For n mod 6 = 0, no formula is currently known.

For (n - 3) mod 6 = 0, T(n,k) = A000096(2n-3)*k^2/4 + A005563(n)*k/2 = (2n^2 - 3n)*k^2/4 + (n^2 + 2n)*k/2.

E.g., T(15,k) = 405k^2/4 + 255k/2.

Conjectured formula for the columns for odd values of k for n>=3:

T(n,k) = A001105((k+1)/2)*n^2 - A051890((k+1)/2)*n + 1 = (k^2 + 2k + 1)*n^2/2 - (k^2 + 3)*n/2 + 1.

E.g., T(n,9) = 50n^2 - 42n + 1.

Conjectured formula for T(n,2):

T(n,2) = 2*A249127(n) = 2*floor(3n/2)*n, for n>=3.

No formula is current known for the columns for even values of k for k>=4.