A007678 -id:A007678 - cp's OEIS frontend

A331450 Irregular triangle read by rows: Take a regular n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., max_k.

1, 4, 10, 0, 1, 18, 6, 35, 7, 7, 0, 1, 56, 24, 90, 36, 18, 9, 0, 0, 1, 120, 90, 10, 176, 132, 44, 22, 276, 168, 377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1, 476, 378, 98, 585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1, 848, 672, 128, 48, 1054, 901, 357, 136, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1404, 954, 72, 18, 18, 1653, 1444, 646, 190, 57, 38, 2200, 1580, 580, 120, 0, 20, 2268, 2520, 903, 462, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jan 25 2020

Computed by Scott R. Shannon, Jan 24 2020
See A331451 for a version of this triangle giving the counts for k = 3 through n.
Mitosis of convex polygons, by Scott R. Shannon and N. J. A. Sloane, Dec 13 2021 (Start)
Borrowing a term from biology, we can think of this process as the "mitosis" of a regular polygon. Row 6 of this triangle shows that a regular hexagon "mitoses" into 18 triangles and 4 quadrilaterals, which we denote by 3^18 4^6.
What if we start with a convex but not necessarily regular n-gon? Let M(n) denote the number of different decompositions into cells that can be obtained. For n = 3, 4, and 5 there is only one possibility. For n = 6 there are two possibilities, 3^18 4^6 and 3^19 4^3 5^3. For n = 7 there are at least 11 possibilities. So the sequence M(n) for n >= 3 begins 1, 1, 1, 2, >=11, ...
The links below give further information. See also A350000. (End)

			A hexagon with all diagonals drawn contains 18 triangles and 6 quadrilaterals, so row 6 is [18, 6].
Triangle begins:
  1,
  4,
  10, 0, 1,
  18, 6,
  35, 7, 7, 0, 1,
  56, 24,
  90, 36, 18, 9, 0, 0, 1,
  120, 90, 10,
  176, 132, 44, 22, 0, 0, 0, 0, 1
  276, 168,
  377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1,
  476, 378, 98,
  585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1,
  848, 672, 128, 48,
  1054, 901, 357, 136, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  1404, 954, 72, 18, 18,
  1653, 1444, 646, 190, 57, 38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
  2200, 1580, 580, 120, 0, 20,
  2268, 2520, 903, 462, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  2992, 2860, 814, 66, 44, 44,
  3749, 2990, 1564, 644, 115, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  ...
The row sums are A007678, the first column is A062361.

M. Rubinstein, Drawings of A007678 for n=4,5,6,....
Scott R. Shannon, Rows 3 through 596 of A331450.
N. J. A. Sloane, Illustration for row n=9 of A331450. [9-gon with one representative for each type of polygonal cell labeled with its number of sides].
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
Scott R. Shannon and N. J. A. Sloane, N. J. A. Sloane, Table showing number of ways to "mitose" a convex n-gon. [From here on the links are related to the mitosis problem, and are in logical rather than alphabetical order]
Scott R. Shannon, Illustration for mitosis 5.1 of a pentagon (the cell counts are the same whether of not the pentagon is regular)
Scott R. Shannon, Illustration for mitosis 6.1 of a hexagon with a triple point (the cell counts are the same whether of not the hexagon is regular, as long as it has a triple point)
Scott R. Shannon, Illustration for mitosis 6.2 of a hexagon (without a triple point)
N. J. A. Sloane, Illustration for mitosis 7a of a 7-gon
Scott R. Shannon, A second (colored) illustration for mitosis 7a of a 7-gon
Scott R. Shannon, Illustration for mitosis 7b
Scott R. Shannon, Illustration for mitosis 7c
Scott R. Shannon, Illustration for mitosis 7d
Scott R. Shannon, Illustration for mitosis 7e
Scott R. Shannon, Illustration for mitosis 7f
Scott R. Shannon, Illustration for mitosis 7g
Scott R. Shannon, Illustration 1 for mitosis 7h
Scott R. Shannon, Illustration 2 for mitosis 7h (same cell counts as preceding illustration but a different polygon - look at the brown cells)
Scott R. Shannon, Illustration for mitosis 7i
Scott R. Shannon, Illustration for mitosis 7j
Scott R. Shannon, Illustration for mitosis 7k
M. F. Hasler, Nine examples of dissections of convex 7-gons [These are all subsumed in the above illustrations]

Cf. A007678, A062361, A331451, A341729, A341730, A335000.

Added "regular" to definition. - N. J. A. Sloane, Mar 06 2021

A331451 Triangle read by rows: Take an n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n.

Original entry on oeis.org

1, 4, 0, 10, 0, 1, 18, 6, 0, 0, 35, 7, 7, 0, 1, 56, 24, 0, 0, 0, 0, 90, 36, 18, 9, 0, 0, 1, 120, 90, 10, 0, 0, 0, 0, 0, 176, 132, 44, 22, 0, 0, 0, 0, 1, 276, 168, 0, 0, 0, 0, 0, 0, 0, 0, 377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1, 476, 378, 98, 0, 0, 0, 0, 0, 0, 0, 0, 0, 585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1, 848, 672, 128, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jan 25 2020

Computed by Scott R. Shannon, Jan 24 2020

			A hexagon with all diagonals drawn contains 18 triangles, 6 quadrilaterals, and no pentagons or hexagons, so row 6 is [18, 6, 0, 0].
Triangle begins:
1,
4,0,
10,0,1,
18,6,0,0,
35,7,7,0,1,
56,24,0,0,0,0,
90,36,18,9,0,0,1,
120,...
The row sums are A007678, the first column is A062361.

M. Rubinstein, Drawings of A007678 for n=4,5,6,...
Scott R. Shannon, Rows 3 through 45
N. J. A. Sloane, Illustration for row n=9. [9-gon with one representative for each type of polygonal cell labeled with its number of sides]

Cf. A007678, A062361.

See A331450 for a version of this triangle in which trailing zeros in the rows have been omitted.

By counting edges in two ways, we have the identity Sum_k k*T(n,k) + n = 2*A135565(n). E.g. for n=7, 3*35+4*7+5*7+6*0+7*1+7 = 182 = 2*A135565(7).

A341729 Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678).

Original entry on oeis.org

3, 3, 5, 4, 7, 4, 9, 5, 11, 4, 13, 5, 15, 6, 17, 7, 19, 8, 21, 8, 23, 6, 25, 8, 27, 8, 29, 6, 31, 7, 33, 8, 35, 8, 37, 8, 39, 12, 41, 8, 43, 10, 45, 8, 47, 10, 49, 10, 51, 8, 53, 10, 55, 10, 57, 10, 59, 9, 61, 10, 63, 10, 65, 9, 67, 10, 69, 10, 71, 10, 73, 10, 75, 12, 77, 10, 79, 10

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Mar 06 2021

nonn

For a(2*n) see A341730.
Theorem: a(2*n+1) = 2*n+1. For the proof see A342222.
It would be nice to have a bigger b-file.

Scott R. Shannon, Table of n, a(n) for n = 3..765

Cf. A007678, A062361, A288177, A331450, A331451, A342222.

Bisections: A144396, A341730.

a(141) and beyond from Scott R. Shannon, Nov 30 2021

A335102 Irregular triangle read by rows: consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.

Original entry on oeis.org

0, 0, 0, 1, 5, 12, 1, 35, 40, 8, 1, 126, 140, 20, 0, 1, 330, 228, 60, 12, 0, 1, 715, 644, 112, 0, 0, 0, 1, 1365, 1168, 208, 0, 0, 0, 0, 1, 2380, 1512, 216, 54, 54, 0, 0, 0, 1, 3876, 3360, 480, 0, 0, 0, 0, 0, 0, 1, 5985, 5280, 660, 0, 0, 0, 0, 0, 0, 0, 1, 8855, 6144, 864, 264, 24, 0, 0, 0, 0, 0, 0, 12, 12650

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 23 2020

			Table begins:
      0;
      0;
      0;
      1;
      5;
     12,    1;
     35;
     40,    8,   1;
    126;
    140,   20,   0,   1;
    330;
    228,   60,  12,   0,   1;
    715;
    644,  112,   0,   0,   0,  1;
   1365;
   1168,  208,   0,   0,   0,  0, 1;
   2380;
   1512,  216,  54,  54,   0,  0, 0, 1;
   3876;
   3360,  480,   0,   0,   0,  0, 0, 0, 1;
   5985;
   5280,  660,   0,   0,   0,  0, 0, 0, 0, 1;
   8855;
   6144,  864, 264,  24,   0,  0, 0, 0, 0, 0, 1;
  12650;
  11284, 1196,   0,   0,   0,  0, 0, 0, 0, 0, 0, 1;
  17550;
  15680, 1568,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 1;
  23751;
  13800, 2250, 420, 180, 120, 30, 0, 0, 0, 0, 0, 0, 0, 1;
  31465;
  28448, 2464,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  40920;
  37264, 2992,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  52360;

B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
Scott R. Shannon, Image of the vertices for n=5.
Scott R. Shannon, Image of the vertices for n=6.
Scott R. Shannon, Image of the vertices for n=8.
Scott R. Shannon, Image of the vertices for n=12.
Scott R. Shannon, Image of the vertices for n=13.
Scott R. Shannon, Image of the vertices for n=16.
Scott R. Shannon, Image of the vertices for n=18.
Scott R. Shannon, Image of the vertices for n=20.
Scott R. Shannon, Image of the vertices for n=24.
Scott R. Shannon, Image of the vertices for n=30.
Index entries for sequences related to stained glass windows

Columns give A292104, A101363 (2n-gon), A101364, A101365.
Row sums give A006561.
Cf. A007569, A007678, A053126, A292105, A333275.

If n = 2t+1 is odd then the n-th row has a single term, T(2t+1, 2t+4) = binomial(2t+1,4) (these values are given in A053126).

A341730 Maximum number of sides in any cell in a regular 2n-gon with all diagonals drawn (cf. A007678).

Original entry on oeis.org

3, 4, 4, 5, 4, 5, 6, 7, 8, 8, 6, 8, 8, 6, 7, 8, 8, 8, 12, 8, 10, 8, 10, 10, 8, 10, 10, 10, 9, 10, 10, 9, 10, 10, 10, 10, 12, 10, 10, 10, 10, 10, 10, 10, 14, 10, 10, 10, 10, 12, 12, 10, 11, 10, 12, 10, 12, 12, 14, 14, 12, 12, 12, 12, 14, 12, 12

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Mar 06 2021

nonn

A bisection of A341729.
a(46) = 14, corresponding to a regular 92-gon, suggesting that this sequence may be unbounded (cf. A342222). (The Pfetsch-Ziegler web page discusses a similar question for polygons defined by grid points.) It would be nice to have a b-file.
For which values of n is a(n) odd (and why)?

M. E. Pfetsch and G. M. Ziegler, Large chambers in a lattice polygon, December 13, 2004;
M. E. Pfetsch and G. M. Ziegler, Large chambers in a lattice polygon [Local copy of webpage, pdf only]

Cf. A007678, A062361, A331450, A331451, A341729, A342222.

A349784 Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678), excluding the central n-sided cell for odd values of n.

Original entry on oeis.org

3, 3, 4, 5, 4, 6, 5, 6, 4, 8, 5, 7, 6, 8, 7, 8, 8, 7, 8, 8, 6, 8, 8, 8, 8, 11, 6, 10, 7, 8, 8, 9, 8, 10, 8, 8, 12, 10, 8, 14, 10, 9, 8, 10, 10, 10, 10, 12, 8, 12, 10, 10, 10, 10, 10, 12, 9, 12, 10, 12, 10, 12, 9, 10, 10, 10, 10, 11, 10, 12, 10, 12, 12, 10, 10, 10, 10, 10, 10, 10, 10, 12, 10

Offset: 4

Scott R. Shannon and N. J. A. Sloane, Nov 30 2021

nonn

As a regular n-gon with an odd number of sides always creates an n-sided cell at its center when all its diagonals are drawn, see A342222, this n-sided cell is not considered for odd n.
Although the behavior of the sequence is unknown as n -> infinity, the data up to n = 765 implies the sequence is possibly bounded. In the range studied the 14-gon is the predominant maximum-sided cell for n > 300.
No n-gon is currently known that produces a cell with 17 sides or 19 sides and above, other than the corresponding central n-sided cell for odd values of n.
See A342222 and A342236 for images of the n-gons.

			a(4) = 3 as a regular 4-gon (square) creates four 3-gons (triangles) when all its diagonals are drawn.
a(5) = 3 as a regular 5-gon (pentagon) creates ten 3-gons when all its diagonals are drawn. Also created is a central 5-gon but this cell is not considered.
a(6) = 4 as a regular 6-gon (hexagon) creates eighteen 3-gons and six 4-gons when all its diagonals are drawn.
a(7) = 5 as a regular 7-gon (heptagon) creates thirty-five 3-gons, seven 4-gons and seven 5-gons when all its diagonals are drawn. Also created is a central 7-gon but this cell is not considered.

Scott R. Shannon, Table of n, a(n) for n = 4..765
Scott R. Shannon, Image showing a close-up of the 18-sided cell in the 708-gon. Zoom in to see the vertices marked as white dots around the 18-gon, shown in violet. The bottom three vertices are extremely close together -- if the image were expanded so that the outer two of these vertices were 1cm away from the inner vertex then the size of the entire 708-gon image would be slightly over 12.9 km in diameter.

Cf. A341729, A342236, A007678, A331450.

A341734 a(n) = A007678(2n)/(2n).

Original entry on oeis.org

0, 1, 4, 10, 22, 37, 68, 106, 137, 225, 310, 376, 538, 685, 716, 1058, 1288, 1471, 1842, 2170, 2327, 2941, 3388, 3734, 4412, 4993, 5444, 6306, 7042, 7391, 8680, 9586, 10289, 11585, 12682, 13628, 15078, 16381, 17440, 19210, 20740, 21899, 24038, 25810, 27245, 29613, 31648, 33418, 35992, 38305

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2021

nonn

This is the number of cells in a 1/(2*n)-th sector of a regular (2*n)-gon with all diagonals drawn. See Rubinstein's illustrations in A007678.

			If we divide a regular hexagon with all diagonals drawn into 6 sectors (or pizza slices), each sector contains three triangles and one quadrilateral (cf. A331450), so a(3) = A007678(6)/6 = 24/6 = 4.

Scott R. Shannon, Table of n, a(n) for n = 1..71

Cf. A007678, A331450, A341735.

Row sums of triangle in A342268.

A341735 a(n) = A007678(2*n+1).

Original entry on oeis.org

0, 1, 11, 50, 154, 375, 781, 1456, 2500, 4029, 6175, 9086, 12926, 17875, 24129, 31900, 41416, 52921, 66675, 82954, 102050, 124271, 149941, 179400, 213004, 251125, 294151, 342486, 396550, 456779, 523625, 597556, 679056, 768625, 866779, 974050, 1090986, 1218151, 1356125

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 07 2021

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A007678.

See A341734 for the other bisection (rescaled).

[div(n*(2*n-1)*(2*n^2-n+5), 6) for n in 0:40] |> println # Bruno Berselli, Mar 08 2021

Table[n (2 n - 1) (2 n^2 - n + 5)/6, {n, 0, 40}] (* Bruno Berselli, Mar 08 2021 *)

From Bruno Berselli, Mar 08 2021: (Start)
G.f.: x*(1 + 6*x + 5*x^2 + 4*x^3)/(1 - x)^5.
a(n) = n*(2*n - 1)*(2*n^2 - n + 5)/6. (End)

More terms from Bruno Berselli, Mar 08 2021

A342268 Irregular triangle read by rows: Take a regular (2n)-sided polygon (n>=2) with all diagonals drawn, as in A007678. Then T(n,k) = (1/(2n))*(number of k-sided polygons in that figure) for k = 3, 4, ..., max_k.

Original entry on oeis.org

1, 3, 1, 7, 3, 12, 9, 1, 23, 14, 34, 27, 7, 53, 42, 8, 3, 78, 53, 4, 1, 1, 110, 79, 29, 6, 0, 1, 136, 130, 37, 3, 2, 3, 184, 154, 35, 3, 184, 154, 35, 3, 297, 273, 76, 34, 4, 1, 389, 264, 48, 15, 449, 403, 153, 46, 7, 547, 497, 163, 69, 9, 3, 679, 519, 207, 59, 5, 2, 759, 717, 268, 71, 22, 5

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Mar 07 2021

This is a version of A331450: take the even-indexed rows and divide by the number of vertices. That is, we only consider one sector (or pizza slice).

			Triangle begins:
1;
3, 1;
7, 3;
12, 9, 1;
23, 14;
34, 27, 7;
53, 42, 8, 3;
78, 53, 4, 1, 1;
110, 79, 29, 6, 0, 1;
136, 130, 37, 3, 2, 2;
184, 154, 35, 3;
242, 195, 81, 15, 4, 1;
297, 273, 76, 34, 4, 1;
389, 264, 48, 15;
449, 403, 153, 46, 7;
547, 497, 163, 69, 9, 3;
679, 519, 207, 59, 5, 2;
759, 717, 268, 71, 22, 5;
900, 819, 329, 100, 16, 5, 0, 0, 0, 1;
1079, 885, 271, 82, 9, 1;
...

Scott R. Shannon, Rows 2 through 70.

Cf. A007678.

Row sums give A341734.

A085611 Difference between A007678(2n)/(2n) and (n-1)^2.

Original entry on oeis.org

0, 0, 0, 1, 6, 12, 32, 57, 73, 144, 210, 255, 394, 516, 520, 833, 1032, 1182, 1518, 1809, 1927, 2500, 2904, 3205, 3836, 4368, 4768, 5577, 6258, 6550, 7780, 8625, 9265, 10496, 11526, 12403, 13782, 15012, 15996, 17689, 19140, 20218, 22274, 23961, 25309, 27588, 29532, 31209, 33688

Offset: 1

Jon Perry, Jul 08 2003

nonn

If we define b(n) by b(n)=local(nr,fn,cn); nr=0; fn=floor(n/2); cn=ceiling(n/2); forstep (i=n,4,-2,nr=nr+(i-2)*fn+(i-4)*cn); nr then a(n) is given by (A007678(2n)-b(2n))/(2n).

This b(n) is given by (n-2)*(2*n^2 - 4*n + 3*(-1)^n - 3)/8 for n > 1. - R. J. Mathar, Oct 18 2013

Cf. A007678, A006533, A000290.

apply( {A085611(n)=A007678(2*n)/(2*n)-(n-1)^2}, [1..40]) \\ M. F. Hasler, Aug 05 2021

a(n) = A007678(2*n)/(2*n) - (n-1)^2. - M. F. Hasler, Aug 06 2021

The last term seemed to be corrupted and has now been deleted. - N. J. A. Sloane, Oct 29 2006

Edited and more terms from M. F. Hasler, Aug 06 2021

cp's OEIS Frontend

A331450 Irregular triangle read by rows: Take a regular n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., max_k.

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A331451 Triangle read by rows: Take an n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., n.

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A341729 Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678).

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A335102 Irregular triangle read by rows: consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.

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A341730 Maximum number of sides in any cell in a regular 2n-gon with all diagonals drawn (cf. A007678).

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A349784 Maximum number of sides in any cell in a regular n-gon with all diagonals drawn (cf. A007678), excluding the central n-sided cell for odd values of n.

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A341734 a(n) = A007678(2*n)/(2*n).

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A341735 a(n) = A007678(2*n+1).

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A342268 Irregular triangle read by rows: Take a regular (2*n)-sided polygon (n>=2) with all diagonals drawn, as in A007678. Then T(n,k) = (1/(2*n))*(number of k-sided polygons in that figure) for k = 3, 4, ..., max_k.

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A085611 Difference between A007678(2n)/(2n) and (n-1)^2.

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A341734 a(n) = A007678(2n)/(2n).

A342268 Irregular triangle read by rows: Take a regular (2n)-sided polygon (n>=2) with all diagonals drawn, as in A007678. Then T(n,k) = (1/(2n))*(number of k-sided polygons in that figure) for k = 3, 4, ..., max_k.