A344866 -id:A344866 - cp's OEIS frontend

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

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

Scott R. Shannon, May 30 2021

nonn

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.

			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.

Scott R. Shannon, Table of n, a(n) for n = 1..100
J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
Scott R. Shannon, Image for n = 3. In this and other images the n-gon vertices are highlighted as white dots while the outer open regions, which are not counted, are darkened. The key for the edge-number coloring is shown at the top-left of the image.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 13.
Scott R. Shannon, Image for n = 14.
Scott R. Shannon, Image for n = 15.
Scott R. Shannon, Image for n = 19.
Scott R. Shannon, Image for n = 21.
Scott R. Shannon, Image for n = 24.
Alexander Sidorenko, Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.

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.

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).

A344899 Number of polygon edges formed when every pair of vertices of a regular n-gon are joined by an infinite line.

Original entry on oeis.org

0, 1, 3, 8, 30, 78, 189, 320, 684, 1010, 1815, 2052, 3978, 4718, 7665, 8576, 13464, 12546, 22059, 23720, 34230, 36542, 50853, 47928, 72900, 76466, 101439, 105560, 137634, 115230, 182745, 188672, 238128, 245378, 305235, 294948, 385614, 395390, 480909, 491840, 592860, 544950, 723303, 737528

Offset: 1

Scott R. Shannon, Jun 02 2021

nonn

See A344857 for other examples and images of the polygons.

			a(3) = 3 as the connected vertices form a triangle with three edges. Six infinite edges between the outer regions are also formed but these are not counted.
a(5) = 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 regions are also formed.

J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
Alexander Sidorenko, Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.

Cf. A344907 (number of edges for odd n), A344857 (number of polygons), A146212 (number of vertices), A344866, A344311, A007678, A331450, A344938.

Bisections: A344907, A347322.

Conjectured formula odd n: a(n) = (n^4 - 7*n^3 + 17*n^2 - 11*n)/4 = (n-1)*n*(n^2-6*n+11)/4.
This formula is correct: see the Sidorenko link. - N. J. A. Sloane, Sep 12 2021
See also A344907.
a(n) = A344857(n) + A146212(n) - 1 (Euler's theorem.).

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

Scott R. Shannon, Jun 03 2021

See A344857 for examples and images of the polygons.

			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;

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).

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

A344311 Number of polygons formed outside a regular n-gon when every pair of vertices of the n-gon are joined by an infinite line.

Original entry on oeis.org

0, 0, 0, 0, 5, 18, 49, 96, 198, 320, 550, 708, 1235, 1582, 2415, 2912, 4284, 4518, 7068, 8080, 11025, 12430, 16445, 16992, 23650, 26000, 32994, 35896, 44863, 42090, 59675, 64064, 77880, 83232, 99960, 102132, 126429, 133950, 157833, 166560, 194750, 191310, 237790, 249480, 287595, 301070

Offset: 1

Scott R. Shannon, Jun 01 2021

nonn

See A344857 for other examples and images of the polygons.

Only finite polygons are counted. - N. J. A. Sloane, Jun 11 2021

			a(5) = 5 as the five connected vertices of a regular pentagon form five triangles outside the pentagon, along with sixteen polygons on the inside.

Cf. A344857 (total number of polygons), A007678 (number of polygons inside the n-gon), A345025 (total number of polygons and open regions), A344938, A344857, A344866, A146213 (number of vertices).

For odd n, a(n) = (2*n^4 - 15*n^3 + 34*n^2 - 21*n)/24 (see A344857).

a(n) = A344857(n) - A007678(n).

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

A345025 Number of regions formed when every pair of vertices of a regular n-gon are joined by an infinite line.

Original entry on oeis.org

1, 2, 7, 16, 36, 72, 141, 232, 424, 630, 1035, 1284, 2172, 2716, 4081, 4848, 7056, 7290, 11439, 12960, 17620, 19712, 26037, 26568, 37176, 40638, 51571, 55832, 69804, 64440, 92505, 98912, 120352, 128146, 154071, 156348, 194436, 205352, 242269, 254920, 298440, 290766, 363867, 380776, 439516

Offset: 1

Scott R. Shannon, Jun 06 2021

nonn

The count of regions includes both the closed bounded polygons and the open unbounded areas surrounding these polygons with two edges that go to infinity.

See A344857 for further examples and images of the regions.

			a(2) = 2 as an infinite line connecting two points cuts space into two unbounded regions.
a(3) = 7 as the three connected points of the 3-gon form one closed triangle along with six outer unbounded areas, seven regions in total.
a(4) = 16 as the four connected points of the 4-gon form four closed triangle inside the square along with twelve outer unbounded areas, sixteen regions in total.

Scott R. Shannon, Image for n = 3. In this and other images the n-gon vertices are highlighted as white dots while the outer open regions are cross-hatched. The key for the edge-number coloring is shown at the top-left of the image. Note the edge count for open areas also includes the two infinite edges.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.

Cf. A344857 (number of polygons), A344311 (number polygons outside the n-gon), A007678 (number polygons inside the n-gon), A002378 (number of open regions for (n-1)-gon), A146212 (number of vertices), A344866, A344938.

Formula for odd n: a(n) = (n^4 - 7*n^3 + 27*n^2 - 29*n + 8)/8 (see A344857).

For n >= 3, a(n) = A344857(n) + A002378(n-1).

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

Scott R. Shannon, Jun 02 2021

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.

			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.

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

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

A347319 a(n) = (2n+1)(n^3-2*n^2+n+1).

Original entry on oeis.org

1, 3, 15, 91, 333, 891, 1963, 3795, 6681, 10963, 17031, 25323, 36325, 50571, 68643, 91171, 118833, 152355, 192511, 240123, 296061, 361243, 436635, 523251, 622153, 734451, 861303, 1003915, 1163541, 1341483, 1539091, 1757763, 1998945, 2264131, 2554863, 2872731, 3219373, 3596475, 4005771

Offset: 0

N. J. A. Sloane, Sep 12 2021

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

A bisection of A146212, analogous to A344866 and A344907.

a(n)=(2*n+1)*(n^3-2*n^2+n+1) \\ Charles R Greathouse IV, Oct 21 2022

def A347319(n): return n*(n**2*(2*n - 3) + 3) + 1 # Chai Wah Wu, Sep 12 2021

From Chai Wah Wu, Sep 12 2021: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (-3*x^4 - 36*x^3 - 10*x^2 + 2*x - 1)/(x - 1)^5. (End)

A347320 A344857(2*n).

Original entry on oeis.org

0, 4, 42, 176, 540, 1152, 2534, 4608, 6984, 12580, 19250, 26016, 39988, 55076, 63570, 97920, 127024, 155088, 203946, 253360, 289044, 378884, 456918, 528672, 648200, 763620, 869022, 1040368, 1204196, 1318440, 1588750, 1812224, 2013132, 2329476, 2626330, 2910672, 3303804, 3687748, 4039854, 4555520, 5043000, 5471088, 6133778, 6740976, 7295760, 8090020, 8836094, 9544128, 10481296, 11384900

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Sep 12 2021, extended Jun 23 2025

No formula is known.

Cf. A344857, A344866.

cp's OEIS Frontend

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

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A344899 Number of polygon edges formed when every pair of vertices of a regular n-gon are joined by an infinite line.

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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.

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A344311 Number of polygons formed outside a regular n-gon when every pair of vertices of the n-gon are joined by an infinite line.

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A345025 Number of regions formed when every pair of vertices of a regular n-gon are joined by an infinite line.

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

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A347319 a(n) = (2*n+1)*(n^3-2*n^2+n+1).

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A347320 A344857(2*n).

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A347319 a(n) = (2n+1)(n^3-2*n^2+n+1).