A332376 -id:A332376 - cp's OEIS frontend

A091908 Number of interior intersection points made by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.

0, 1, 12, 13, 48, 49, 108, 109, 192, 193, 300, 301, 432, 433, 576, 589, 768, 769, 972, 961, 1200, 1201, 1452, 1405, 1728, 1729, 2028, 2029, 2352, 2341, 2700, 2701, 3072, 3073, 3444, 3469, 3888, 3889, 4332, 4297, 4800, 4777, 5292, 5293, 5724, 5809, 6348

Offset: 1

Hugo Pfoertner, Feb 19 2004

nonn

In a drawing the distinction between simple and multiple intersection points may be difficult due to near-coincidences. E.g. there are no coincident intersections for n=7.
Note that 3 divides a(2k)-1 and a(2k+1). - T. D. Noe, Jun 29 2005
The interior intersection points only can be the result of the concurrency of 2 or 3 segments by construction. It is easy to see that the total number of 2-intersections N2 is 3*(n-1)^2 (which includes every 3-intersection as two 2-intersections) by symmetry. But we are interested in excluding the concurrency of more than 2. By Ceva's theorem necessary and sufficient condition for 3 concurrent segments that connect the edges with the opposite side, the number of 3-intersections N3 is the same as the number of (i,j,k) belonging to [1,n-1]x[1,n-1]x[1,n-1] such that (i/(n-i))*(j/(n-j))*(k/(n-k))=1. Thus the terms a(n)=N2-2*N3. - Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006
If n is even then a(n) < 3*(n-1)^2; if n is odd then a(n) = 3*(n-1)^2 except for n in A332378. - N. J. A. Sloane, Feb 14 2020

			a(3)=12 because the 3*2 line segments intersect each other in 12 distinct points (see pictures given at link)
a(4)=13 because the 27 intersections form 6 two line intersection points and 7 three line intersection points.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000
Hugo Pfoertner, Visualization of diagonal intersections in an equilateral triangle.
Index entries for sequences formed by drawing all diagonals in regular polygon

Cf. A091910 = radial locations of intersection points, A092098 = number of regions that the line segments cut the triangle into, A006561.

For the basic properties of the underlying graph, see A092098 (cells), A331782 (vertices), A331782 (vertices), A332376 & A332377 (edges). - N. J. A. Sloane, Feb 14 2020

for(n=1,70,conc=0;for(i=1,n-1,for(j=1,n-1,for(k=1,n-1,if(i*j*k/((n-i)*(n-j)*(n-k))==1,conc++))));print1(3*(n-1)^2-2*conc,",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006

More terms from T. D. Noe, Jun 29 2005

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006

A331782 Total number of vertices in graph formed by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.

Original entry on oeis.org

3, 7, 21, 25, 63, 67, 129, 133, 219, 223, 333, 337, 471, 475, 621, 637, 819, 823, 1029, 1021, 1263, 1267, 1521, 1477, 1803, 1807, 2109, 2113, 2439, 2431, 2793, 2797, 3171, 3175, 3549, 3577, 3999, 4003, 4449, 4417, 4923, 4903, 5421, 5425, 5859, 5947, 6489, 6397, 7059, 7063, 7653, 7657, 8271, 8275, 8889

Offset: 1

N. J. A. Sloane, Feb 13 2020

nonn

a(n) <= 3*(n^2-n+1), with equality iff n is odd and not a member of A332378. A331423 gives the difference between a(n) and the upper bound.

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Cf. A091908, A092098 (number of cells), A332376 (number of edges), A332378, A331423.

a(n) = A091908(n) + 3*n.

A357008 Number of edges in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

Original entry on oeis.org

3, 9, 27, 57, 99, 135, 219, 297, 351, 489, 603, 645, 867, 1017, 1107, 1353, 1539, 1575, 1947, 2127, 2295, 2649, 2907, 3021, 3459, 3753, 3855, 4359, 4707, 4821, 5403, 5769, 5967, 6537, 6897, 6957, 7779, 8217, 8451, 9003, 9603, 9837, 10587, 11061, 11211, 12153, 12699, 12897, 13827, 14409, 14715

Offset: 0

Scott R. Shannon, Sep 08 2022

nonn

See A356984 and A357007 for images of the triangles.

Scott R. Shannon, Table of n, a(n) for n = 0..250

Cf. A356984 (regions), A357007 (vertices), A274586, A332376, A333027, A344896.

a(n) = A356984(n) + A357007(n) - 1 by Euler's formula.

Conjecture: a(n) = 6*n^2 + 3 for equilateral triangles that only contain simple vertices when cut by n internal equilateral triangles. This is never the case if (n + 1) mod 3 = 0 for n > 3.

A335412 a(n) is the number of edges formed by n-secting the angles of an equilateral triangle.

Original entry on oeis.org

3, 12, 39, 54, 123, 144, 255, 282, 435, 432, 663, 702, 939, 984, 1263, 1314, 1635, 1692, 2055, 2082, 2523, 2592, 3039, 3114, 3603, 3684, 4215, 4302, 4875, 4932, 5583, 5682, 6339, 6444, 7143, 7254, 7995, 8112, 8895, 8982, 9843, 9972, 10839, 10974, 11883, 12024

Offset: 1

Lars Blomberg, Jun 08 2020

nonn

See A277402 for illustrations.

Lars Blomberg, Table of n, a(n) for n = 1..500

Cf. A332376, A277402 (regions), A335411 (vertices), A335413 (ngons).

Empirically for 12 < n < 500: a(n) = a(n-2) + a(n-10) - a(n-12) + 240.
Conjectures from Colin Barker, Jun 08 2020: (Start)
G.f.: 3*x*(1 + 3*x + 8*x^2 + 2*x^3 + 14*x^4 + 2*x^5 + 14*x^6 + 2*x^7 + 14*x^8 - 10*x^9 + 25*x^10 + 11*x^11 - 6*x^12) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-10) - a(n-11) - a(n-12) + a(n-13) for n>13.
(End)
Colin Barker's recurrence conjecture holds for 13 < n <= 500. Lars Blomberg, Jun 12 2020

A332377 One-third of total number of edges in graph formed by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces.

Original entry on oeis.org

1, 4, 13, 18, 41, 48, 85, 94, 145, 156, 221, 234, 313, 328, 415, 438, 545, 564, 685, 700, 841, 864, 1013, 1014, 1201, 1228, 1405, 1434, 1625, 1650, 1861, 1894, 2113, 2148, 2369, 2418, 2665, 2704, 2965, 2988, 3281, 3312, 3613, 3658, 3919, 4008, 4325

Offset: 1

N. J. A. Sloane, Feb 13 2020

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Cf. A091908, A092098 (number of cells), A331782 (number of vertices), A332376.

a(n) = (A092098(n) + A331782(n) - 1)/3 = A332376(n)/3.

A342152 The number of edges on a vesica piscis formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

2, 8, 42, 148, 438, 936, 2010, 3462, 6038, 8816, 14606, 20504, 29854, 39790, 54618, 70142, 92662, 115718, 147494, 177500, 223506, 267872, 326142, 384274, 460302, 535896, 631886, 726674, 848126, 965592, 1115194, 1259926, 1440558, 1616940, 1833130, 2042602, 2300498, 2549756, 2851626, 3139854

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 02 2021

nonn

The terms are from numeric computation - no formula for a(n) is currently known.

See A341877 for images of the regions and A341878 for images of the vertices.

Wikipedia, Vesica piscis.

Cf. A341877 (regions), A341878 (vertices), A342153 (n-gons), A135565, A332376, A340613, A340687.

a(n) = A341877(n) + A341878(n) - 1.

A356119 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in an equilateral triangle when straight line segments connect the three corner vertices to the points dividing the sides into n equal parts.

Original entry on oeis.org

1, 6, 12, 3, 3, 1, 24, 6, 36, 9, 9, 7, 48, 24, 6, 72, 21, 15, 19, 84, 48, 12, 6, 108, 51, 33, 25, 132, 78, 18, 18, 168, 69, 51, 43, 180, 120, 48, 18, 216, 135, 57, 61, 252, 156, 66, 36, 294, 159, 105, 67, 312, 234, 84, 48, 372, 225, 117, 103, 408, 264, 138, 60, 456, 291, 159, 121, 486, 372, 138, 84

Offset: 1

Scott R. Shannon, Jul 27 2022

The maximum sided k-gon up to n = 250 is the 6-gon; it is likely this is the maximum sided k-gon for all n.

See A092098 for more images of the triangle.

			The table begins:
1;
6;
12,  3,   3,   1;
24,  6;
36,  9,   9,   7;
48,  24,  6;
72,  21,  15,  19;
84,  48,  12,  6;
108, 51,  33,  25;
132, 78,  18,  18;
168, 69,  51,  43;
180, 120, 48,  18;
216, 135, 57,  61;
252, 156, 66,  36;
294, 159, 105, 67;
312, 234, 84,  48;
372, 225, 117, 103;
408, 264, 138, 60;
456, 291, 159, 121;
486, 372, 138, 84;
.
.
See the attached text file for more examples.

Scott R. Shannon, Table for n = 1..250.
Scott R. Shannon, Image for n = 20.

Cf. A092098 (regions), vertices (A331782), edges (A332376), A335413.

Sum of row(n) = A092098(n).

cp's OEIS Frontend

A091908 Number of interior intersection points made by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.

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A331782 Total number of vertices in graph formed by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces, counting coinciding intersection points only once.

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A357008 Number of edges in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

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A335412 a(n) is the number of edges formed by n-secting the angles of an equilateral triangle.

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A332377 One-third of total number of edges in graph formed by the straight line segments connecting the edges of an equilateral triangle with the n-1 points resulting from a subdivision of the sides into n equal pieces.

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A342152 The number of edges on a vesica piscis formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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A356119 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in an equilateral triangle when straight line segments connect the three corner vertices to the points dividing the sides into n equal parts.

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