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.

0, 0, 0, 1, 5, 15, 35, 70, 124, 200, 300, 445, 627, 875, 1189, 1564, 2006, 2568, 3225, 4035, 4972, 6030, 7250, 8701, 10323, 12156, 14235, 16554, 19124, 22072, 25250, 28863, 32827, 37166, 41949, 47142, 52653, 58794, 65503, 72741, 80437

Offset: 1

Author

N. J. A. Sloane, Aug 05 2017

Keywords

• nonn

Comments

Only intersection points above the line are counted.

a(n) <= binomial(n,4) (A000332), since that is the number of pairs of intersecting semicircles. See A290461 for the differences.

The first time a triple intersection occurs is for n=9. Two fourfold intersections occur for n=13. - Torsten Sillke, Jul 27 2017

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

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

References

• Torsten Sillke, email to N. J. A. Sloane, Jul 27 2017 (giving values for a(1)-a(13)).

Links

• David Applegate, Table of n, a(n) for n = 1..500
• Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
• M. F. Hasler, Illustration for a(9) = 124. (First instance where a triple intersection occurs, whence a(9) < binomial(9,4).)
• M. F. Hasler, Illustration for a(9) = 124 [Another version, showing baseline]
• M. F. Hasler, Interactive web page for drawing the illustration for a(n).
• Torsten Sillke, Illustration for a(13) = 627
• N. J. A. Sloane, Illustration for a(5) = 5.
• N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
• N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
• Zahlenjagd, Winter 2010 Problem (asks for a(10)).

Crossrefs

See A006561 for an analogous problem on a circle.

Cf. A290461, A290465, A290726.

See A290865, A290866, A290867, A290876, A332723 for further properties of these configurations.

Programs

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

Python

from itertools import combinations
from fractions import Fraction
def A290447(n):
    p,p2 = set(), set()
    for b,c,d in combinations(range(1,n),3):
        e = b + d - c
        f1, f2, g = Fraction(b*d,e), Fraction(b*d*(c-b)*(d-c),e**2), (n-1)*e - 2*b*d
        for i in range(n-d):
            if 2*i*e < g:
                p2.add((i+f1, f2))
            elif 2*i*e == g:
                p.add(f2)
            else:
                break
    return len(p)+2*len(p2) # Chai Wah Wu, Aug 08 2017

Extensions

More terms from David Applegate, Aug 07 2017