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Number of regions in an equilateral triangular figure formed by the straight line segments connecting all vertices and all points that divide the sides into n equal parts.
+10
62
1, 12, 75, 252, 715, 1572, 3109, 5676, 9291, 14556, 22081, 32502, 44935, 62868, 83286, 108384, 140152, 181710, 225565, 282978, 342792, 415614, 502318, 606642, 708505, 839874, 983007, 1141416, 1315102, 1529526, 1733476, 1994550, 2259420, 2559990, 2878053, 3237414, 3593521, 4047906, 4510590, 5002350, 5506918, 6128100, 6704800, 7414518, 8113992, 8858622, 9682927, 10626774, 11478142, 12519492
EXAMPLE
a(2)=12 because the 6 line segments mutually connecting the vertices and the mid-side nodes form 12 congruent right triangles of two different sizes.
a(3)=75: 48 triangles, 24 quadrilaterals and 3 pentagons are formed. See pictures at Pfoertner link.
CROSSREFS
Cf. A092866 (number of intersections), A274585 (number of points both inside and on the triangle sides), A274586 (number of edges), A331911 (number of n-gons). Cf. A092098 (regions in triangle cut by line segments connecting vertices with subdivision points on opposite side), A006533 (regions formed by all diagonals in regular n-gon), A002717 (triangles in triangular matchstick arrangement).
Number of edges formed by sides and straight "chords" in a right triangle when each side is divided by vertices into n equal segments.
+10
27
3, 21, 132, 429, 1272, 2826, 5640, 10461, 17094, 26847, 41046, 61041, 84051, 118974, 157209, 204393, 264855, 346524, 428880, 541683, 654087, 793611, 961179, 1167468, 1357515, 1615209, 1891980, 2198019, 2530275, 2957808, 3341439, 3860652, 4371006, 4959636, 5572167, 6277722, 6950064, 7859406, 8763780, 9722571, 10687506, 11934912, 13029834, 14450598, 15805026, 17250795, 18863397, 20763204, 22372839, 24450474
Number of intersections inside an equilateral triangular figure formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts. If three or more lines meet at an interior point this intersection is counted only once.
+10
26
0, 4, 49, 166, 543, 1237, 2511, 4762, 7777, 12262, 18933, 28504, 39078, 56065, 73879, 95962, 124653, 164761, 203259, 258646, 311233, 377932, 458793, 560755, 648936, 775258, 908893, 1056520, 1215087, 1428193, 1607871, 1866007, 2111488, 2399545, 2694010, 3040201, 3356433, 3811387, 4253074, 4720102, 5180466, 5806687, 6324906, 7035949, 7690900, 8392036, 9180330, 10136287, 10894551, 11930833
COMMENTS
A detailed example for n=5 is given at the Pfoertner link.
EXAMPLE
a(2)=4 because there are 3 intersection points between the triangle medians and the line segments connecting the midpoints of the sides plus the intersection of the 3 medians at the centroid.
MAPLE
Inter:= proc(p1x, p1y, p2x, p2y, q1x, q1y, q2x, q2y)
local det, x, y;
det:= p1x*q1y-p1x*q2y-p1y*q1x+p1y*q2x-p2x*q1y+p2x*q2y+p2y*q1x-p2y*q2x;
if det = 0 then return NULL fi;
x:= (p1x*p2y*q1x-p1x*p2y*q2x-p1x*q1x*q2y+p1x*q1y*q2x-p1y*p2x*q1x+p1y*p2x*q2x+p2x*q1x*q2y-p2x*q1y*q2x)/det;
y:= (p1x*p2y*q1y-p1x*p2y*q2y-p1y*p2x*q1y+p1y*p2x*q2y-p1y*q1x*q2y+p1y*q1y*q2x+p2y*q1x*q2y-p2y*q1y*q2x)/det;
if x >0 and y > 0 and x + y < 1 then [x, y]
else NULL
fi
end proc:
F:= proc(n) local A, B, C, Pairs, Pts;
A:= [seq([j/n, 0], j=0..n)];
B:= [seq([0, j/n], j=0..n)];
C:= [seq([j/n, 1-j/n], j=0..n)];
Pairs:= [seq(seq([A[i], B[j]], i=2..n+1), j=2..n+1),
seq(seq([A[i], C[j]], i=1..n), j=1..n),
seq(seq([B[i], C[j]], i=1..n), j=2..n+1)];
Pts:= {seq(seq(Inter(op(Pairs[i][1]), op(Pairs[i][2]), op(Pairs[j][1]), op(Pairs[j][2])), j=1..i-1), i=2..nops(Pairs))};
nops(Pts);
end proc:
MATHEMATICA
Inter[{p1x_, p1y_}, {p2x_, p2y_}, {q1x_, q1y_}, {q2x_, q2y_}] := Module[ {det, x, y}, det = p1x q1y - p1x q2y - p1y q1x + p1y q2x - p2x q1y + p2x q2y + p2y q1x - p2y q2x; If[det == 0, Return[Nothing]]; x = (p1x p2y q1x - p1x p2y q2x - p1x q1x q2y + p1x q1y q2x - p1y p2x q1x + p1y p2x q2x + p2x q1x q2y - p2x q1y q2x)/det; y = (p1x p2y q1y - p1x p2y q2y - p1y p2x q1y + p1y p2x q2y - p1y q1x q2y + p1y q1y q2x + p2y q1x q2y - p2y q1y q2x)/det; If[x > 0 && y > 0 && x + y < 1, {x, y}, Nothing]];
F[n_] := F[n] = Module[{A, B, K, Pairs, Pts}, A = Table[{j/n, 0}, {j, 0, n}]; B = Table[{0, j/n}, {j, 0, n}]; K = Table[{j/n, 1 - j/n}, {j, 0, n}]; Pairs = {Table[Table[{A[[i]], B[[j]]}, {i, 2, n+1}], {j, 2, n+1}], Table[Table[{A[[i]], K[[j]]}, {i, 1, n}], {j, 1, n}], Table[Table[ {B[[i]], K[[j]]}, {i, 1, n}], {j, 2, n+1}]} // Flatten[#, 2]&; Pts = Table[Table[Inter[Pairs[[i, 1]], Pairs[[i, 2]], Pairs[[j, 1]], Pairs[[j, 2]]], {j, 1, i-1}], {i, 2, Length[Pairs]}]; Flatten[Pts, 1] // Union // Length];
CROSSREFS
Cf. A092867 (regions formed by the diagonals), A274585 (points both inside and on the triangle sides), A274586 (edges). Cf. A006561 (number of intersections of diagonals of regular n-gon), A091908 (intersections between line segments connecting vertices with subdivision points on opposite side).
Place n points in general position on each side of an equilateral triangle, and join every pair of the 3*n+3 boundary points by a chord; sequence gives number of regions in the resulting planar graph.
+10
8
1, 13, 82, 307, 841, 1891, 3718, 6637, 11017, 17281, 25906, 37423, 52417, 71527, 95446, 124921, 160753, 203797, 254962, 315211, 385561, 467083, 560902, 668197, 790201, 928201, 1083538, 1257607, 1451857, 1667791, 1906966, 2170993, 2461537, 2780317, 3129106, 3509731, 3924073, 4374067
COMMENTS
"In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet.
FORMULA
Conjecture: a(n) = (1/4)*(9*n^4 + 12*n^3 + 15*n^2 + 12*n + 4).
Place n points in general position on each side of an equilateral triangle, and join every pair of the 3*n+3 boundary points by a chord; sequence gives number of edges in the resulting planar graph.
+10
7
3, 24, 153, 588, 1635, 3708, 7329, 13128, 21843, 34320, 51513, 74484, 104403, 142548, 190305, 249168, 320739, 406728, 508953, 629340, 769923, 932844, 1120353, 1334808, 1578675, 1854528, 2165049, 2513028, 2901363, 3333060, 3811233, 4339104, 4920003, 5557368, 6254745, 7015788
COMMENTS
"In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet.
FORMULA
Conjecture: a(n) = (3/2)*(3*n^4 + 4*n^3 + 3*n^2 + 4*n + 2).
a(n) is the total number of points (both boundary and interior points) in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter), where the interior points are counted with multiplicity.
+10
6
5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728, 211445, 287266, 381855, 498080, 639013, 807930, 1008311, 1243840, 1518405, 1836098, 2201215, 2618256, 3091925, 3627130, 4228983, 4902800, 5654101, 6488610, 7412255, 8431168, 9551685, 10780346, 12123895, 13589280, 15183653, 16914370, 18788991
COMMENTS
An equivalent definition: Place n-1 points in general position on each side of a square, and join every pair of the 4*n+4 boundary points by a chord; sequence gives number of vertices in the resulting planar graph. "In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet. - Scott R. Shannon and N. J. A. Sloane, Nov 05 2023 Equivalently, this is A334697(n) + 4*n.
FORMULA
Theorem: a(n) = n*(17*n^3-30*n^2+19*n+4)/2.
G.f.: x*(5 + 33*x + 135*x^2 + 31*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)
MATHEMATICA
LinearRecurrence[{5, -10, 10, -5, 1}, {5, 58, 375, 1376, 3685}, 50] (* Paolo Xausa, Nov 14 2023 *)
PROG
(PARI) Vec(x*(5 + 33*x + 135*x^2 + 31*x^3) / (1 - x)^5 + O(x^40)) \\ Colin Barker, May 31 2020
Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of vertices in the resulting planar graph.
+10
3
3, 12, 5, 72, 58, 10, 282, 375, 185, 19, 795, 1376, 1155, 451, 42, 1818, 3685, 4090, 2734, 938, 57, 3612, 8130, 10700, 9478, 5523, 1711, 135, 6492, 15743, 23235, 24463, 18858, 9981, 2943, 171, 10827, 27760, 44485, 52639, 48230, 33771, 16740, 4646, 341
COMMENTS
"In general position" implies that the internal lines (or chords) formed from the n*k edge points only have simple intersections; there is no interior points where three or more such chords meet. Note that for even-n n-gons, with n>=6, the chords from the n corner points do create non-simple intersections.
FORMULA
T(3,k) = A367117(k) = (9/4)*k^4 + 3*k^3 + (3/4)*k^2 + 3*k + 3. Conjectured:
T(4,k) = A334698(k+1) = (17/2)*k^4 + 19*k^3 + (31/2)*k^2 + 10*k + 5. T(5,k) = (45/2)*k^4 + 60*k^3 + 60*k^2 + (65/2)*k + 10.
T(6,k) = (195/4)*k^4 + (285/2)*k^3 + (627/4)*k^2 + 84*k + 19.
T(7,k) = (371/4)*k^4 + 287*k^3 + (1337/4)*k^2 + 182*k + 42.
T(8,k) = 161*k^4 + 518*k^3 + 627*k^2 + 348*k + 57.
T(9,k) = 261*k^4 + 864*k^3 + (2151/2)*k^2 + (1215/2)*k + 135.
T(10,k) = (1605/4)*k^4 + (2715/2)*k^3 + (6905/4)*k^2 + 990*k + 171.
EXAMPLE
The table begins:
3, 12, 72, 282, 795, 1818, 3612, 6492, 10827, 17040, 25608, 37062, 51987,...
5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728,...
10, 185, 1155, 4090, 10700, 23235, 44485, 77780, 126990, 196525, 291335,...
19, 451, 2734, 9478, 24463, 52639, 100126, 174214, 283363, 437203, 646534,...
42, 938, 5523, 18858, 48230, 103152, 195363, 338828, 549738, 846510, 1249787,...
57, 1711, 9981, 33771, 85849, 182847, 345261, 597451, 967641, 1487919, 2194237,...
135, 2943, 16740, 56106, 141885, 301185, 567378, 980100, 1585251, 2434995,...
171, 4646, 26336, 87831, 221351, 468746, 881496, 1520711, 2457131, 3771126,...
341, 7128, 39666, 131450, 330165, 697686, 1310078, 2257596, 3644685, 5589980,...
313, 10204, 57199, 189214, 474361, 1000948, 1877479, 3232654, 5215369, 7994716,...
728, 14677, 80457, 264602, 661570, 1393743, 2611427, 4492852, 7244172,...
771, 19909, 109586, 359892, 898591, 1891121, 3540594, 6087796, 9811187,...
1380, 27030, 146565, 479370, 1194600, 2511180, 4697805, 8072940, 13004820,...
1393, 35085, 191353, 625477, 1557297, 3271213, 6116185, 10505733,......
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