A277402 -id:A277402 - cp's OEIS frontend

A335413 Irregular table read by rows: T(n,k) = number of k-sided polygons formed by n-secting the angles of an equilateral triangle for k >= 3.

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, 126, 90, 12, 6, 168, 69, 51, 43, 180, 120, 48, 18, 216, 123, 81, 49, 240, 180, 54, 36, 288, 171, 99, 73, 312, 234, 84, 48, 372, 225, 117, 103, 396, 288, 126, 60

Offset: 1

Lars Blomberg, Jun 08 2020

See A277402 for illustrations.

			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;
126, 90, 12, 6;
168, 69, 51, 43;
180, 120, 48, 18;
216, 123, 81, 49;
240, 180, 54, 36;
288, 171, 99, 73;
312, 234, 84, 48;

Lars Blomberg, Table of n, a(n) for n = 1..1991

Cf. A277402 (regions), A335411 (vertices), A335412 (edges).

A335411 a(n) is the number of vertices formed by n-secting the angles of an equilateral triangle.

Original entry on oeis.org

3, 7, 21, 25, 63, 67, 129, 133, 219, 199, 333, 337, 471, 475, 633, 637, 819, 823, 1029, 1009, 1263, 1267, 1521, 1525, 1803, 1807, 2109, 2113, 2439, 2419, 2793, 2797, 3171, 3175, 3573, 3577, 3999, 4003, 4449, 4429, 4923, 4927, 5421, 5425, 5943, 5947, 6489

Offset: 1

Lars Blomberg, Jun 08 2020

nonn

See A277402 for illustrations.

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

Cf. A331782, A277402 (regions), A335412 (edges), A335413 (ngons).

Empirically for 12 < n < 500: a(n) = a(n-2) + a(n-10) - a(n-12) + 120.
Conjectures from Colin Barker, Jun 08 2020: (Start)
G.f.: x*(3 + 4*x + 11*x^2 + 24*x^4 + 24*x^6 + 24*x^8 - 24*x^9 + 45*x^10 + 20*x^11 - 11*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
Empirical: a(2*k - 1) = 3*(4*k^2 - 6*k + 3), for k >= 1. - Ivan N. Ianakiev, Jul 15 2020

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

A278823 4-Portolan numbers: number of regions formed by n-secting the angles of a square.

Original entry on oeis.org

1, 4, 29, 32, 93, 84, 189, 188, 321, 316, 489, 460, 693, 676, 933, 916, 1205, 1180, 1505, 1496, 1849, 1836, 2229, 2188, 2645, 2616, 3097, 3060, 3577, 3536, 4089, 4064, 4645, 4604, 5237, 5176, 5857, 5808, 6513, 6472, 7201, 7160, 7933, 7900, 8693, 8648, 9497

Offset: 1

Ethan Beihl, Nov 28 2016

nonn

m-Portolan numbers for m>3 (especially m even) are more difficult than m=3 (A277402) because Ceva's theorem doesn't immediately give us a condition for redundant intersections. The values for n <= 23 were found by brute force in Mathematica, as follows:
1. Solve for the coordinates of all intersections between lines within the square, recording multiplicity.
2. Use an elementary Euler's-formula method as in Poonen and Rubinstein 1998 to turn the intersection-count into a region-count.

			For n=3, the 4*(3-1) = 8 lines intersect to make 12 triangles, 8 kites, 8 irregular quadrilaterals, and an octagon in the middle. The total number of regions a(3) is therefore 12+8+8+1 = 29.

Lars Blomberg, Table of n, a(n) for n = 1..500
Ethan Beihl, Pictures for some small n
Lars Blomberg, Coloured illustration for n=4
Lars Blomberg, Coloured illustration for n=5
Lars Blomberg, Coloured illustration for n=64
Lars Blomberg, Coloured illustration for n=65
B. Poonen and M. Rubinstein (1998) The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11(1), pp. 135-156, doi:10.1137/S0895480195281246, arXiv:math.MG/9508209 (typos corrected)

3-Portolan numbers (equilateral triangle): A277402.
n-sected sides (not angles): A108914.
Cf. A277402, A335526 (vertices), A335527 (edges), A335528 (ngons).

For n = 2k - 1, a(n) is close to 18k^2 - 26k + 9. For n = 2k, a(n) is close to 18k^2 - 26k + 12. The residuals are related to the structure of redundant intersections in the figure.

a(24) and beyond from Lars Blomberg, Jun 12 2020

cp's OEIS Frontend

A335413 Irregular table read by rows: T(n,k) = number of k-sided polygons formed by n-secting the angles of an equilateral triangle for k >= 3.

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A335411 a(n) is the number of vertices formed by n-secting the angles of an equilateral triangle.

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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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A278823 4-Portolan numbers: number of regions formed by n-secting the angles of a square.

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