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.

Showing 1-4 of 4 results.

A357058 Number of regions in a square when n internal squares are drawn between the 4n points that divide each side into n+1 equal parts.

Original entry on oeis.org

1, 5, 17, 37, 65, 93, 145, 181, 257, 309, 401, 457, 577, 653, 785, 869, 1025, 1109, 1297, 1413, 1601, 1725, 1937, 2041, 2305, 2453, 2705, 2861, 3137, 3289, 3601, 3765, 4089, 4293, 4625, 4801, 5185, 5405, 5769, 5993, 6401, 6605, 7057, 7309, 7737, 8013, 8465, 8673, 9217, 9477, 9993, 10309
Offset: 0

Views

Author

Scott R. Shannon, Sep 10 2022

Keywords

Comments

The even values of n that yield squares with non-simple intersections are 32, 38, 44, 50, 54, 62, 76, 90, 98, ... .

Links

Crossrefs

Cf. A357060 (vertices), A357061 (edges), A108914, A355838, A355798, A356984 (triangle).

Formula

a(n) = A357061(n) - A357060 (n) + 1 by Euler's formula.
Conjecture: a(n) = 4*n^2 + 1 for squares that only contain simple intersections when cut by n internal squares. This is never the case for odd n >= 5.

A357061 Number of edges in a square when n internal squares are drawn between the 4n points that divide each side into n+1 equal parts.

Original entry on oeis.org

4, 12, 36, 76, 132, 180, 292, 348, 516, 604, 804, 892, 1156, 1284, 1572, 1708, 2052, 2180, 2596, 2796, 3204, 3412, 3876, 4012, 4612, 4860, 5412, 5668, 6276, 6508, 7204, 7460, 8172, 8524, 9252, 9516, 10372, 10740, 11532, 11900, 12804, 13100, 14116, 14532, 15468, 15940, 16932, 17196, 18436
Offset: 0

Views

Author

Scott R. Shannon, Sep 10 2022

Keywords

Comments

The even values of n that yield squares with non-simple intersections are 32, 38, 44, 50, 54, 62, 76, 90, 98, ... .
See A357058 and A357060 for images of the squares.

Crossrefs

Cf. A357058 (regions), A357060 (vertices), A355948, A355840, A355800, A357008 (triangle).

Formula

a(n) = A357058(n) + A357060(n) - 1 by Euler's formula.
Conjecture: a(n) = 8*n^2 + 4 for squares that only contain simple intersections when cut by n internal squares. This is never the case for odd n >= 5.

A357197 Number of vertices in a hexagon when n internal hexagons are drawn between the 6n points that divide each side into n+1 equal parts.

Original entry on oeis.org

6, 12, 30, 60, 102, 156, 222, 300, 390, 468, 606, 708, 870, 1020, 1152, 1356, 1542, 1740, 1950, 2112, 2406, 2652, 2910, 3072, 3462, 3756, 4062, 4350, 4710, 4974, 5406, 5772, 6126, 6540, 6918, 7260, 7782, 8220, 8646, 8946, 9606, 10032, 10590, 11052, 11568, 12156, 12702, 13116, 13830, 14388
Offset: 0

Views

Author

Scott R. Shannon, Sep 17 2022

Keywords

Comments

Unlike similar dissections of the triangle and square, see A357007 and A357060, there is no obvious pattern for n values that yield hexagons with non-simple intersections; these n values begin 9, 11, 14, 19, 23, 27, 29, 32, 34, 35, 38, 39, 41, 43, ... .

Links

Crossrefs

Cf. A357196 (regions), A357198 (edges), A330846, A357007 (triangle), A357060 (square).

Formula

a(n) = A357198(n) - A357196(n) + 1 by Euler's formula.
Conjecture: a(n) = 6*n^2 + 6 for hexagons that only contain simple intersections when cut by n internal hexagons.

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.

Original entry on oeis.org

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

Views

Author

Scott R. Shannon, Sep 19 2022

Keywords

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

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.
Showing 1-4 of 4 results.

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