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Search: id:A091908
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| A091908 |
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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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+0
5
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| 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
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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 (noe(AT)sspectra.com), 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
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LINKS
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Hugo Pfoertner, Visualization of diagonal intersections in an equilateral triangle.
Sequences formed by drawing all diagonals in regular polygon
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EXAMPLE
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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.
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PROGRAM
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(PARI) 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
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CROSSREFS
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Cf. A091910 radial locations of intersection points, A092098 number of regions that the line segments cut the triangle into, A006561.
Sequence in context: A041298 A041689 A041296 this_sequence A041300 A041302 A041304
Adjacent sequences: A091905 A091906 A091907 this_sequence A091909 A091910 A091911
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KEYWORD
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nonn
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AUTHOR
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Hugo Pfoertner (hugo(AT)pfoertner.org), Feb 19 2004
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EXTENSIONS
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More terms from T. D. Noe (noe(AT)sspectra.com), Jun 29 2005
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 26 2006
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