A357235 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of vertices in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

3, 6, 4, 15, 8, 5, 30, 20, 10, 6, 51, 40, 25, 12, 7, 66, 68, 50, 30, 14, 8, 111, 88, 85, 60, 35, 16, 9, 150, 148, 130, 102, 70, 40, 18, 10, 171, 168, 185, 156, 119, 80, 45, 20, 11, 246, 260, 250, 222, 182, 136, 90, 50, 22, 12, 303, 296, 325, 300, 259, 208, 153, 100, 55, 24, 13

Offset: 3

Author

Scott R. Shannon, Sep 19 2022

Keywords

• nonn
• tabl

Comments

Conjecture: the only n-gons that contain non-simple intersections are the 3-gon (triangle), 4-gon (square), and 6-gon (hexagon).

Examples

			The table begins:
   3,  6, 15,  30,  51,  66, 111, 150, 171,  246,  303,  312,  435,  510,  543, ...
   4,  8, 20,  40,  68,  88, 148, 168, 260,  296,  404,  436,  580,  632,  788, ...
   5, 10, 25,  50,  85, 130, 185, 250, 325,  410,  505,  610,  725,  850,  985, ...
   6, 12, 30,  60, 102, 156, 222, 300, 390,  468,  606,  708,  870, 1020, 1152, ...
   7, 14, 35,  70, 119, 182, 259, 350, 455,  574,  707,  854, 1015, 1190, 1379, ...
   8, 16, 40,  80, 136, 208, 296, 400, 520,  656,  808,  976, 1160, 1360, 1576, ...
   9, 18, 45,  90, 153, 234, 333, 450, 585,  738,  909, 1098, 1305, 1530, 1773, ...
  10, 20, 50, 100, 170, 260, 370, 500, 650,  820, 1010, 1220, 1450, 1700, 1970, ...
  11, 22, 55, 110, 187, 286, 407, 550, 715,  902, 1111, 1342, 1595, 1870, 2167, ...
  12, 24, 60, 120, 204, 312, 444, 600, 780,  984, 1212, 1464, 1740, 2040, 2364, ...
  13, 26, 65, 130, 221, 338, 481, 650, 845, 1066, 1313, 1586, 1885, 2210, 2561, ...
  14, 28, 70, 140, 238, 364, 518, 700, 910, 1148, 1414, 1708, 2030, 2380, 2758, ...
  15, 30, 75, 150, 255, 390, 555, 750, 975, 1230, 1515, 1830, 2175, 2550, 2955, ...
See the attached text file for further examples.
See A357007, A357060, A357197 for more images of the n-gons.
		

Links

• Scott R. Shannon, Extended table for n = 3..50, k = 0..75.
• Scott R. Shannon, Image of T(5,20) = 2005.
• Scott R. Shannon, Image of T(7,10) = 707.

Crossrefs

Cf. A357216 (regions), A357254 (edges), A357007 (triangle), A357060 (square), A357197 (hexagon), A007569, A146212.

Formula

T(n,k) = A357254(n,k) - A357216(n,k) + 1 by Euler's formula.

T(n,0) = n.

T(n,1) = 2n.

Conjectured formula for all columns for n >= 7: T(n,k) = n*k^2 + n.

T(3,k) = A357007(k).

T(4,k) = A357060(k).

T(6,k) = A357197(k).

Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = n*k^2 + n.