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A290447 Consider n equally spaced points along a line and join every pair of points by a semicircle above the line; a(n) is the number of intersection points.

%I A290447 #78 May 21 2021 07:09:43
%S A290447 0,0,0,1,5,15,35,70,124,200,300,445,627,875,1189,1564,2006,2568,3225,
%T A290447 4035,4972,6030,7250,8701,10323,12156,14235,16554,19124,22072,25250,
%U A290447 28863,32827,37166,41949,47142,52653,58794,65503,72741,80437
%N A290447 Consider n equally spaced points along a line and join every pair of points by a semicircle above the line; a(n) is the number of intersection points.
%C A290447 Only intersection points above the line are counted.
%C A290447 a(n) <= binomial(n,4) (A000332), since that is the number of pairs of intersecting semicircles. See A290461 for the differences.
%C A290447 The first time a triple intersection occurs is for n=9. Two fourfold intersections occur for n=13. - Torsten Sillke, Jul 27 2017
%C A290447 If the line is the x-axis and the two semicircles are for (x_1,0),(x_2,0) and (x_3,0),(x_4,0) (with x_1 < x_2, x_3 < x_4, and x_1 < x_3) then they intersect if and only if x_1 < x_3 < x_2 < x_4, and the intersection point has coordinates (x,y) with x=(x_3*x_4 - x_1*x_2) / (x_3 + x_4 - x_1 - x_2) and y^2 = (x_3-x_1)*(x_4-x_1)*(x_2-x_3)*(x_4-x_2) / (x_3 + x_4 - x_1 - x_2)^2.  This allows identification of distinct (and duplicate) intersection points using only rational arithmetic. - _David Applegate_, Aug 07 2017
%C A290447 Suppose x_i are integers in the range 0 <= x_i < n. Then (x,y) is an intersection point if and only if (n-1-x,y) is an intersection point. Suppose x_4 < n-1. If (x,y) is an intersection point, then (i+x,y) is an intersection point for i = 1,..,n-1-x_4. - _Chai Wah Wu_, Aug 09 2017
%D A290447 Torsten Sillke, email to _N. J. A. Sloane_, Jul 27 2017 (giving values for a(1)-a(13)).
%H A290447 David Applegate, <a href="/A290447/b290447.txt">Table of n, a(n) for n = 1..500</a>
%H A290447 Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, <a href="http://neilsloane.com/doc/rose_5.pdf">Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids</a>, (2021). Also arXiv:2009.07918.
%H A290447 M. F. Hasler, <a href="/A290447/a290447_1.svg">Illustration for a(9) = 124</a>. (First instance where a triple intersection occurs, whence a(9) < binomial(9,4).)
%H A290447 M. F. Hasler, <a href="/A290447/a290447_2.svg">Illustration for a(9) = 124</a> [Another version, showing baseline]
%H A290447 M. F. Hasler, <a href="/A290447/a290447.html">Interactive web page for drawing the illustration for a(n).</a>
%H A290447 Torsten Sillke, <a href="/A290447/a290447.ps">Illustration for a(13) = 627</a>
%H A290447 N. J. A. Sloane, <a href="/A290447/a290447.png">Illustration for a(5) = 5</a>.
%H A290447 N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, <a href="https://vimeo.com/237029685">Part I</a>, <a href="https://vimeo.com/237030304">Part 2</a>, <a href="https://oeis.org/A290447/a290447_slides.pdf">Slides.</a> (Mentions this sequence)
%H A290447 N. J. A. Sloane (in collaboration with Scott R. Shannon), <a href="/A331452/a331452.pdf">Art and Sequences</a>, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
%H A290447 Zahlenjagd, <a href="http://www.zahlenjagd.at/aufgaben.php">Winter 2010 Problem</a> (asks for a(10)).
%o A290447 (PARI) A290447(n,U=[])={for(A=1,n-3,for(C=A+1,n-2,for(B=C+1,n-1,for(D=B+1,n,U=setunion(U,[[(C*D-A*B)/(C+D-A-B),(C-A)*(D-A)*(C-B)*(D-B)/(C+D-A-B)^2]])))));#U} \\ _M. F. Hasler_, Aug 07 2017
%o A290447 (Python)
%o A290447 from itertools import combinations
%o A290447 from fractions import Fraction
%o A290447 def A290447(n):
%o A290447     p,p2 = set(), set()
%o A290447     for b,c,d in combinations(range(1,n),3):
%o A290447         e = b + d - c
%o A290447         f1, f2, g = Fraction(b*d,e), Fraction(b*d*(c-b)*(d-c),e**2), (n-1)*e - 2*b*d
%o A290447         for i in range(n-d):
%o A290447             if 2*i*e < g:
%o A290447                 p2.add((i+f1, f2))
%o A290447             elif 2*i*e == g:
%o A290447                 p.add(f2)
%o A290447             else:
%o A290447                 break
%o A290447     return len(p)+2*len(p2) # _Chai Wah Wu_, Aug 08 2017
%Y A290447 See A006561 for an analogous problem on a circle.
%Y A290447 Cf. A290461, A290465, A290726.
%Y A290447 See A290865, A290866, A290867, A290876, A332723 for further properties of these configurations.
%K A290447 nonn
%O A290447 1,5
%A A290447 _N. J. A. Sloane_, Aug 05 2017
%E A290447 More terms from _David Applegate_, Aug 07 2017