A357235 -id:A357235 - cp's OEIS frontend

A357216 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of regions 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.

1, 4, 1, 13, 5, 1, 28, 17, 6, 1, 49, 37, 21, 7, 1, 70, 65, 46, 25, 8, 1, 109, 93, 81, 55, 29, 9, 1, 148, 145, 126, 97, 64, 33, 10, 1, 181, 181, 181, 151, 113, 73, 37, 11, 1, 244, 257, 246, 217, 176, 129, 82, 41, 12, 1, 301, 309, 321, 295, 253, 201, 145, 91, 45, 13, 1

Offset: 3

Scott R. Shannon, Sep 18 2022

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

			The table begins:
  1,  4, 13,  28,  49,  70, 109, 148, 181,  244,  301,  334,  433,  508,  565, ...
  1,  5, 17,  37,  65,  93, 145, 181, 257,  309,  401,  457,  577,  653,  785, ...
  1,  6, 21,  46,  81, 126, 181, 246, 321,  406,  501,  606,  721,  846,  981, ...
  1,  7, 25,  55,  97, 151, 217, 295, 385,  475,  601,  715,  865, 1015, 1159, ...
  1,  8, 29,  64, 113, 176, 253, 344, 449,  568,  701,  848, 1009, 1184, 1373, ...
  1,  9, 33,  73, 129, 201, 289, 393, 513,  649,  801,  969, 1153, 1353, 1569, ...
  1, 10, 37,  82, 145, 226, 325, 442, 577,  730,  901, 1090, 1297, 1522, 1765, ...
  1, 11, 41,  91, 161, 251, 361, 491, 641,  811, 1001, 1211, 1441, 1691, 1961, ...
  1, 12, 45, 100, 177, 276, 397, 540, 705,  892, 1101, 1332, 1585, 1860, 2157, ...
  1, 13, 49, 109, 193, 301, 433, 589, 769,  973, 1201, 1453, 1729, 2029, 2353, ...
  1, 14, 53, 118, 209, 326, 469, 638, 833, 1054, 1301, 1574, 1873, 2198, 2549, ...
  1, 15, 57, 127, 225, 351, 505, 687, 897, 1135, 1401, 1695, 2017, 2367, 2745, ...
  1, 16, 61, 136, 241, 376, 541, 736, 961, 1216, 1501, 1816, 2161, 2536, 2941, ...
  ...
See the attached text file for further examples.
See A356984, A357058, A357196 for more images of the n-gons.

Scott R. Shannon, Extended table for n = 3..50, k = 0..75.
Scott R. Shannon, Image of T(5,20) = 2001.
Scott R. Shannon, Image of T(7,10) = 701.

Cf. A357235 (vertices), A357254 (edges), A356984 (triangle), A357058 (square), A357196 (hexagon), A007678, A344857.

T(n,k) = A357254(n,k) - A357235(n,k) + 1 by Euler's formula.
T(n,0) = 1.
T(n,1) = n + 1.
Conjectured formula for all columns for n >= 7: T(n,k) = n*k^2 + 1.
T(3,k) = A356984(k).
T(4,k) = A357058(k).
T(6,k) = A357196(k).
Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = n*k^2 + 1.

A357254 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges 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.

Original entry on oeis.org

3, 9, 4, 27, 12, 5, 57, 36, 15, 6, 99, 76, 45, 18, 7, 135, 132, 95, 54, 21, 8, 219, 180, 165, 114, 63, 24, 9, 297, 292, 255, 198, 133, 72, 27, 10, 351, 348, 365, 306, 231, 152, 81, 30, 11, 489, 516, 495, 438, 357, 264, 171, 90, 33, 12, 603, 604, 645, 594, 511, 408, 297, 190, 99, 36, 13

Offset: 3

Scott R. Shannon, Sep 20 2022

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

			The table begins:
   3,  9,  27,  57,  99, 135,  219,  297,  351,  489,  603,  645,  867, 1017, ...
   4, 12,  36,  76, 132, 180,  292,  348,  516,  604,  804,  892, 1156, 1284, ...
   5, 15,  45,  95, 165, 255,  365,  495,  645,  815, 1005, 1215, 1445, 1695, ...
   6, 18,  54, 114, 198, 306,  438,  594,  774,  942, 1206, 1422, 1734, 2034, ...
   7, 21,  63, 133, 231, 357,  511,  693,  903, 1141, 1407, 1701, 2023, 2373, ...
   8, 24,  72, 152, 264, 408,  584,  792, 1032, 1304, 1608, 1944, 2312, 2712, ...
   9, 27,  81, 171, 297, 459,  657,  891, 1161, 1467, 1809, 2187, 2601, 3051, ...
  10, 30,  90, 190, 330, 510,  730,  990, 1290, 1630, 2010, 2430, 2890, 3390, ...
  11, 33,  99, 209, 363, 561,  803, 1089, 1419, 1793, 2211, 2673, 3179, 3729, ...
  12, 36, 108, 228, 396, 612,  876, 1188, 1548, 1956, 2412, 2916, 3468, 4068, ...
  13, 39, 117, 247, 429, 663,  949, 1287, 1677, 2119, 2613, 3159, 3757, 4407, ...
  14, 42, 126, 266, 462, 714, 1022, 1386, 1806, 2282, 2814, 3402, 4046, 4746, ...
  15, 45, 135, 285, 495, 765, 1095, 1485, 1935, 2445, 3015, 3645, 4335, 5085, ...
  ...
See the attached text file for further examples.
See A356984, A357058, A357196 for images of the n-gons.

Scott R. Shannon, Extended table for n = 3..50, k = 0..75.

Cf. A357216 (regions), A357235 (vertices), A357008 (triangle), A357061 (square), A357198 (hexagon), A356984, A357058, A357196, A135565, A344899.

T(n,k) = A357216(n,k) + A357235(n,k) - 1 by Euler's formula.
T(n,0) = n.
T(n,1) = 3n.
Conjectured formula for all columns for n >= 7: T(n,k) = 2n*k^2 + n.
T(3,k) = A357008(k).
T(4,k) = A357061(k).
T(6,k) = A357198(k).
Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = 2n*k^2 + n.

cp's OEIS Frontend

A357216 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of regions 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.

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