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 1-10 of 12 results. Next

A347322 A344899(2*n).

Original entry on oeis.org

1, 8, 78, 320, 1010, 2052, 4718, 8576, 12546, 23720, 36542, 47928, 76466, 105560, 115230, 188672, 245378, 294948, 395390, 491840, 544950, 737528
Offset: 1

Views

Author

N. J. A. Sloane, Sep 12 2021

Keywords

Comments

No formula is known.

Crossrefs

Cf. A344899, A344907.

A344857 Number of polygons formed when every pair of vertices of a regular n-gon are joined by an infinite line.

Original entry on oeis.org

0, 0, 1, 4, 16, 42, 99, 176, 352, 540, 925, 1152, 2016, 2534, 3871, 4608, 6784, 6984, 11097, 12580, 17200, 19250, 25531, 26016, 36576, 39988, 50869, 55076, 68992, 63570, 91575, 97920, 119296, 127024, 152881, 155088, 193104, 203946, 240787, 253360, 296800, 289044, 362061, 378884, 437536, 456918
Offset: 1

Views

Author

Scott R. Shannon, May 30 2021

Keywords

Comments

For odd n, a(n) is given by the equation in the Formula section below. See also A344866. For even n no such equation is currently known, although one similar to the general formula for the number of polygons inside an n-gon, see A007678, likely exists.
The number of open regions, those outside the polygons with unbounded area and two edges that go to infinity, for n >= 3 is given by n*(n-1) = A002378(n-1).
See A345025 for the total number of all areas, both polygons and open regions.

Examples

			a(1) = a(2) = 0 as no polygon can be formed by one or two connected points.
a(3) = 1 as the connected vertices form a triangle, while the six regions outside the triangle are open.
a(4) = 4 as the four connected vertices form four triangles inside the square. Twelve open regions outside these polygons are also formed.
a(5) = 16 as the five connected vertices form eleven polygons inside the regular pentagon while also forming five triangles outside the pentagon, giving sixteen polygons in total. Twenty open regions outside these polygons are also formed.
a(6) = 42 as the six connected vertices form twenty-four polygons inside the regular hexagon while also forming eighteen polygons outside the hexagon, giving forty-two polygons in total. Thirty open regions outside these polygons are also formed.
See the linked images above for further examples.

Links

Crossrefs

Cf. A344311 (number of finite regions outside the n-gon), A007678 (number inside the n-gon), A345025 (total number of regions), A344866 (number for odd n), A146212 (number of vertices), A344899 (number of edges), A344938 (number of k-gons), A002378 (number of open regions for (n-1)-gon).
Bisections: A344866, A347320.

Formula

For odd n, a(n) = (n^4 - 7*n^3 + 19*n^2 - 21*n + 8)/8 = (n-1)^2*(n^2-5*n+8)/8. This was conjectured by Scott R. Shannon and proved by Alexander Sidorenko on Sep 10 2021 (see link). - N. J. A. Sloane, Sep 12 2021
See also A344866.
a(n) = A344311(n) + A007678(n).

A146212 Number of intersection points of all lines through pairs of vertices of a regular n-gon.

Original entry on oeis.org

3, 5, 15, 37, 91, 145, 333, 471, 891, 901, 1963, 2185, 3795, 3969, 6681, 5563, 10963, 11141, 17031, 17293, 25323, 21913, 36325, 36479, 50571, 50485, 68643, 51661, 91171, 90753, 118833, 118355, 152355, 139861, 192511, 191445, 240123, 238481
Offset: 3

Views

Author

T. D. Noe, Oct 28 2008

Keywords

Comments

This includes intersection points outside of the n-gon. Note that for odd n, n divides a(n); for even n, n divides a(n)-1. For odd n, it appears that a(n)=n*(n^3-7*n^2+15*n-1)/8.
That formula for odd n is correct: see the Sidorenko link. - N. J. A. Sloane, Sep 12 2021

Examples

			a(5)=15 because there are 5 points inside the pentagon, 5 points on the pentagon and five points outside of the pentagon.

Links

Crossrefs

Cf. A006561, A007569, A146213.
Bisection: A347319, A347321.

Formula

There is a formula for odd n: see Comment section and the Sidorenko link. - N. J. A. Sloane, Sep 12 2021

Extensions

More terms from Jon E. Schoenfield, Nov 10 2008
Definition clarified by N. J. A. Sloane, Jun 06 2025

A358783 The number of edges formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

2, 12, 16, 120, 138, 532, 432, 1548, 1450, 3696, 3120, 7488, 6874, 13560, 12000, 23256, 20970, 36328, 32400, 54852, 51194, 79672, 70752, 113100, 105274, 154764, 144480, 206712, 195810, 272800, 255808, 352176, 335002, 446740, 422784, 560920, 534698, 695760, 659440, 850340, 815682
Offset: 2

Views

Author

Scott R. Shannon, Nov 30 2022

Keywords

Comments

Conjecture: for odd values of n all vertices are simple, other than those defining the diameters of the circles. No formula for n, or only the odd values of n, is currently known.
See A358746 and A358782 for images of the circles.
The author thanks Zach Shannon some of whose code was used in the generation of this sequence.
If n is odd, the circle containing the initial n points is not part of the graph (compare A370976-A370979). - N. J. A. Sloane, Mar 25 2024

Crossrefs

Cf. A358746 (vertices), A358782 (regions), A359009 (k-gons), A135565, A344899.
See allso A370976-A370979.

Formula

a(n) = A358746(n) + A358782(n) - 1 by Euler's formula.

A344866 Number of polygons formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

Original entry on oeis.org

0, 1, 16, 99, 352, 925, 2016, 3871, 6784, 11097, 17200, 25531, 36576, 50869, 68992, 91575, 119296, 152881, 193104, 240787, 296800, 362061, 437536, 524239, 623232, 735625, 862576, 1005291, 1165024, 1343077, 1540800, 1759591, 2000896, 2266209, 2557072, 2875075, 3221856, 3599101, 4008544, 4451967
Offset: 1

Views

Author

Scott R. Shannon, Jun 01 2021

Keywords

Comments

This is the odd-indexed subsequence of A344857. See A344857 for images of the polygons.

Examples

			a(3) = 16 as the five connected vertices form eleven polygons inside the regular pentagon while also forming five triangles outside the pentagon, giving sixteen polygons in total.

Links

Crossrefs

Cf. A344857 (number for even and odd n), A344311, A344938, A007678, A341735 (number inside the n-gon), A344899 (number of edges).
See also A347320.

Programs

  • Mathematica
    LinearRecurrence[{5,-10,10,-5,1},{0,1,16,99,352},40] (* Harvey P. Dale, Jun 01 2025 *)
  • Python
    def A344866(n): return n*(n*(n*(2*n - 11) + 23) - 21) + 7 # Chai Wah Wu, Sep 12 2021

Formula

a(n) = 2*n^4 - 11*n^3 + 23*n^2 - 21*n + 7.
G.f.: x^2*(1 + 11*x + 29*x^2 + 7*x^3)/(1 - x)^5. - Stefano Spezia, Jun 04 2021

Extensions

Edited by N. J. A. Sloane, Sep 12 2021

A344938 Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when every pair of vertices of a regular n-gon are joined by an infinite line, for k = 3, 4, ..., max_k.

Original entry on oeis.org

1, 4, 15, 0, 1, 36, 6, 70, 21, 7, 0, 1, 112, 64, 189, 108, 36, 18, 0, 0, 1, 270, 220, 50, 407, 352, 110, 55, 0, 0, 0, 0, 1, 624, 528, 884, 689, 325, 91, 0, 26, 0, 0, 0, 0, 1, 1162, 1092, 266, 14, 1530, 1545, 480, 270, 45, 0, 0, 0, 0, 0, 0, 0, 1, 2080, 2032, 416, 80
Offset: 3

Views

Author

Scott R. Shannon, Jun 03 2021

Keywords

Comments

See A344857 for examples and images of the polygons.

Examples

			A pentagon with all vertices connected forms 10 triangles inside the pentagon, 5 triangles outside the pentagon, giving 15 triangles in all, and 1 smaller pentagon inside the pentagon, so row 3 is [15,0,1].
The table begins:
1;
4;
15,0,1;
36,6;
70,21,7,0,1;
112,64;
189,108,36,18,0,0,1;
270,220,50;
407,352,110,55,0,0,0,0,1;
624,528;
884,689,325,91,0,26,0,0,0,0,1;
1162,1092,266,14;
1530,1545,480,270,45,0,0,0,0,0,0,0,1;
2080,2032,416,80;
2567,2754,1003,374,17,68,0,0,0,0,0,0,0,0,1;
3402,3366,180,18,18;
3952,4807,1672,475,95,76,0,19,0,0,0,0,0,0,0,0,1;
5380,5360,1580,240,0,20;
5943,7392,2583,1260,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
7590,9020,2310,132,132,66;
9430,9775,4508,1518,253,46,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
11304,12288,2280,144;
13025,14650,6250,2375,200,75,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
16042,16952,5954,728,260,52;
17064,22464,7884,2700,567,189,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
21616,24192,7056,2016,168,28;
23751,29319,11281,3828,348,319,0,87,29,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
29880,29010,4140,540;
30814,39370,15314,5177,341,496,0,62,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
37440,42624,14240,3008,544,64;
41481,49335,19305,7854,891,363,66,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;

Crossrefs

Cf. A344857 (total number of polygons), A344899 (number of edges), A146212 (number of vertices), A344866, A344311, A007678, A331450 (number of k-gons inside the regular n-gon).

Formula

Sum of row(n) = A344857(n) = A344311(n) + A007678(n).

A344896 Number of polygon edges formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

Original entry on oeis.org

3, 21, 186, 705, 2196, 5046, 10212, 19149, 31524, 50049, 77100, 114633, 159453, 225978, 298833, 390069, 508137, 663426, 824868, 1040727, 1258833, 1532565, 1859331, 2254938, 2628309, 3130227, 3664920, 4261503, 4917939, 5742696, 6503331, 7511280, 8507322, 9658806, 10855989, 12237066
Offset: 1

Views

Author

Scott R. Shannon, Jun 24 2021

Keywords

Comments

See A344279 for images of the polygons.

Crossrefs

Cf. A344279 (number of polygons), A344657 (number of vertices), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A344899, A345650.

Formula

a(n) = A344279(n) + A344657(n) - 1.

A351045 Irregular table read by rows: row n gives the number of edges with k facing edges for a regular n-gon with all diagonals drawn, with n>=3 and k>=2.

Original entry on oeis.org

3, 4, 0, 4, 5, 0, 10, 0, 5, 6, 0, 18, 12, 6, 7, 0, 28, 14, 21, 14, 7, 8, 0, 56, 48, 24, 9, 0, 54, 54, 72, 72, 18, 0, 9, 10, 0, 80, 160, 120, 20, 11, 0, 88, 154, 198, 198, 55, 0, 0, 0, 11, 12, 0, 240, 336, 168, 13, 0, 130, 260, 507, 390, 91, 104, 0, 0, 0, 0, 13, 14, 0, 266, 616, 644, 140, 42
Offset: 3

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 30 2022

Keywords

Comments

The number of facing edges for a given edge is the number of other edges in the one (for edges on the outside of the n-gon) or two polygons that the edge forms a part of. For example, for an edge shared between two adjoined triangles the number of facing edges is four, as it faces two edges in each of the two triangles it forms a part of.
All edges that are on the outside of the n-gon have two facing edges as any such edge belongs to only one (interior) triangle. Thus T(n,2) = n. For odd n the central created n-gon, see A342222, is surrounded by triangles, thus the edges that form this central n-gon have (n-1)+(3-1) = n+1 facing edges, thus T(n,n+1) >= n.
For all n-gons with even n, or odd n if the central n-gon is ignored, the maximum k for which row(n,k) > 0 is unknown, although it is clearly related to the maximum sided cell for all n-gons; see A349784.

Examples

			A hexagon with all diagonals drawn has six edges (those on the outside of the hexagon) which form one side of a single triangle and thus face two edges, eighteen edges that adjoin two triangles and thus face four edges, twelve edges that adjoin a triangle and a quadrilateral and thus face five edges, and six edges that adjoin two quadrilaterals and thus face six edges. Thus the row for n = 6 is [6, 0, 18, 12, 6]. See the attached image.
The table begins:
3;
4, 0, 4;
5, 0, 10, 0, 5;
6, 0, 18, 12, 6;
7, 0, 28, 14, 21, 14, 7;
8, 0, 56, 48, 24;
9, 0, 54, 54, 72, 72, 18, 0, 9;
10, 0, 80, 160, 120, 20;
11, 0, 88, 154, 198, 198, 55, 0, 0, 0, 11;
12, 0, 240, 336, 168;
13, 0, 130, 260, 507, 390, 91, 104, 0, 0, 0, 0, 13;
14, 0, 266, 616, 644, 140, 42;
15, 0, 180, 600, 945, 630, 435, 0, 15, 0, 0, 0, 0, 0, 15;
16, 0, 448, 1056, 960, 576, 32;
17, 0, 238, 816, 1853, 1224, 425, 272, 34, 0, 0, 0, 0, 0, 0, 0, 17;
18, 0, 900, 1836, 1314, 108, 144;
19, 0, 304, 1520, 2717, 2128, 798, 304, 95, 0, 19, 0, 0, 0, 0, 0, 0, 0, 19;
20, 0, 1000, 2120, 3280, 1600, 100, 240;
21, 0, 378, 2352, 4494, 3276, 1365, 252, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21;
22, 0, 1056, 3828, 5258, 1716, 374, 396, 132;
.
.
See the linked file for the table n = 3..100.

Links

Crossrefs

Cf. A135565, A342222, A349784, A344899, A342236, A342222, A342236.

Formula

Sum of row n = A135565(n).
T(n,2) = n.
T(n,n+1) >= n for odd n.

A344907 Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

Original entry on oeis.org

0, 3, 30, 189, 684, 1815, 3978, 7665, 13464, 22059, 34230, 50853, 72900, 101439, 137634, 182745, 238128, 305235, 385614, 480909, 592860, 723303, 874170, 1047489, 1245384, 1470075, 1723878, 2009205, 2328564, 2684559, 3079890, 3517353, 3999840, 4530339, 5111934, 5747805, 6441228, 7195575
Offset: 1

Views

Author

Scott R. Shannon, Jun 02 2021

Keywords

Comments

This sequences gives the number of polygon edges formed when connecting every pair of vertices of a regular polygon, with an odd number of vertices, by an infinite line.
A bisection of A344899. - N. J. A. Sloane, Sep 12 2021
See A344857 for other examples and images of the polygons.

Examples

			a(3) = 30 as the five connected vertices form a pentagon with fives lines along the pentagon's edges, fifteen lines inside forming eleven polygons, and ten lines outside forming another five triangles. In all these sixteen polygons form thirty edges. Twenty infinite edges between the outer unbounded regions are also formed.

Links

Crossrefs

Cf. A344899 (number of edges for all n-gons), A344866 (number of polygon), A146212, A344857, A344311, A007678, A331450, A344938.
See also A347322.

Programs

  • Python
    def A344907(n): return n*(n*(n*(4*n - 22) + 44) - 35) + 9 # Chai Wah Wu, Sep 12 2021

Formula

a(n) = 4*n^4 - 22*n^3 + 44*n^2 - 35*n + 9 (see Sidorenko link in A344857 for proof).
From Stefano Spezia, Jun 10 2021: (Start)
G.f.: 3*x^2*(1 + 5*x + 23*x^2 + 3*x^3)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)

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

Views

Author

Scott R. Shannon, Sep 20 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,  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.

Links

Crossrefs

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

Formula

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.
Showing 1-10 of 12 results. Next

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