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A092866 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. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A detailed example for n=5 is given at the Pfoertner link.
LINKS
Jessica Gonzalez, Illustration of a(4)=166
Cynthia Miaina Rasamimanananivo and Max Alekseyev, Sage program for this sequence
FORMULA
a(n) = A274585(n) - 3n.
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:
map(F, [$1..20]); # Robert Israel, Jun 30 2016
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];
Table[Print[n, " ", F[n]]; F[n], {n, 1, 20}] (* Jean-François Alcover, Apr 11 2019, after Robert Israel *)
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).
If the boundary points are in general position, we get A367117, A213827, A367118, A367119. - N. J. A. Sloane, Nov 09 2023
Sequence in context: A112533 A016874 A360883 * A340662 A041065 A166838
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Mar 10 2004
EXTENSIONS
a(1) = 0 prepended by Max Alekseyev, Jun 29 2016
a(4) corrected and a(6)-a(20) added by Cynthia Miaina Rasamimanananivo, Jun 28 2016
a(20) corrected by Robert Israel, Jun 30 2016
a(21)-a(50) from Cynthia Miaina Rasamimanananivo, Jun 30 - Aug 23, 2016
"Equilateral" added to definition by N. J. A. Sloane, May 13 2020
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)