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

3,1

Comments

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

Links

Table of n, a(n) for n=3..68.

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.

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.

Example

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.

Crossrefs

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

Sequence in context: A299211 A067979 A091808 * A307460 A128719 A145691

Adjacent sequences: A357232 A357233 A357234 * A357236 A357237 A357238

Keyword

nonn,tabl

Author

Scott R. Shannon, Sep 19 2022

Status

approved