cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A354544 Table read by antidiagonals: T(n,k) (n >= 3, k >= 1) is the number of vertices formed in a regular n-gon by straight line segments when connecting the n corner vertices to the points dividing the sides into k equal parts.

Original entry on oeis.org

3, 7, 5, 21, 25, 10, 25, 81, 61, 19, 63, 157, 285, 205, 42, 67, 301, 476, 541, 358, 57, 129, 381, 1020, 1327, 1526, 681, 135, 133, 665, 1311, 2185, 2682, 2417, 1234, 171, 219, 821, 2215, 3067, 5250, 5073, 4716, 2131, 341, 223, 1109, 2666, 4921, 7246, 8937, 8623, 6861, 3169, 313
Offset: 3

Views

Author

Scott R. Shannon, Aug 18 2022

Keywords

Examples

			The table begins:
3,    7,     21,     25,     63,     67,     129,    133,     219,     223,...
5,    25,    81,     157,    301,    381,    665,    821,     1109,    1353,...
10,   61,    285,    476,    1020,   1311,   2215,   2666,    3810,    4421,...
19,   205,   541,    1327,   2185,   3067,   4921,   6739,    8401,    10507,...
42,   358,   1526,   2682,   5250,   7246,   11214,  14050,   19418,   23094,...
57,   681,   2417,   5073,   8937,   13089,  19473,  26049,   33769,   42497,...
135,  1234,  4716,   8623,   16173,  22780,  34398,  43813,   59391,   71614,...
171,  2131,  6861,   14271,  24731,  36161,  53071,  70751,   91761,   115001,...
341,  3169,  11451,  21143,  38665,  55221,  81983,  105403,  141405,  171689,...
313,  4837,  14653,  31009,  53989,  78589,  115909, 154105,  199777,  249961,...
728,  6787,  23491,  43850,  78858,  113517, 166829, 215788,  287404,  350663,...
771,  9927,  30479,  61951,  105155, 157851, 224043, 299727,  386807,  485367,...
1380, 12856, 43080,  81136,  144180, 208636, 304500, 395536,  524040,  641656,...
1393, 17633, 54001,  109265, 184785, 277745, 392737, 525169,  677729,  849249,...
2397, 22288, 73066,  138177, 243355, 353686, 513264, 668815,  882793,  1083564,...
1855, 27595, 88291,  177085, 302167, 450469, 641539, 855829,  1106119, 1384183,...
3895, 36139, 116337, 220933, 386403, 563351, 814093, 1063393, 1399407, 1721059,...
.
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See the attached text file for more examples and the cross references for further images.

Links

Crossrefs

Cf. A356044 (number of regions), A007569 (first column), A331782 (first row), A355949 (second row).

Formula

T(n,1) = A007569(n).
T(3,k) = A331782(k).
T(4,k) = A355949(k).

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.

Original entry on oeis.org

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

Views

Author

Scott R. Shannon, Aug 28 2022

Keywords

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

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.
Showing 1-2 of 2 results.

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