A290866 -id:A290866 - cp's OEIS frontend

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

N. J. A. Sloane, Aug 05 2017

nonn

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

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

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)).

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.

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

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

More terms from David Applegate, Aug 07 2017

A290865 a(n) = number of regions in the configuration A290447(n).

Original entry on oeis.org

0, 1, 3, 7, 15, 30, 56, 98, 161, 250, 370, 536, 748, 1027, 1379, 1807, 2320, 2954, 3702, 4604, 5652, 6852, 8239, 9858, 11683, 13748, 16086, 18700, 21604, 24887, 28471, 32491, 36907, 41751, 47080, 52876, 59105, 65965, 73440, 81521, 90176

Offset: 1

David Applegate, Aug 12 2017

nonn

			With 3 points, there are 3 semicircles above the baseline, which bound a(3) = 3 regions. With 4 points, there are 6 semicircles, defining 7 regions (use the Halser webpage with n = 3 and 4). - _N. J. A. Sloane_, Aug 12 2017

David Applegate, Table of n, a(n) for n = 1..100
M. F. Hasler, Interactive web page for drawing the illustration for a(n).
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.

Cf. A290447, A290866, A290867, A332723 (number of regions with k edges).

A359254 Number of edges among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

Original entry on oeis.org

4, 26, 96, 216, 480, 882, 1564, 2464, 3804, 5502, 7912, 10864, 14816, 19526, 25508, 32392, 40992, 50818, 62852, 76432, 92460, 110590, 131904, 155306, 182316, 212188, 246316, 283586, 326100

Offset: 2

Scott R. Shannon, Dec 22 2022

A circle is constructed for every pair of the n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed for n points is A001859(n-1).

See A359252 and A359253 for images of the circles.

Cf. A359252 (vertices), A359253 (regions), A359258 (k-gons), A001859, A290866, A359047, A358783.

a(n) = A359252(n) + A359253(n) - 1 by Euler's formula.

A290867 Irregular triangle read by rows: the number of points that are the intersections of k semicircles in the configuration A290447(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 5, 0, 15, 0, 35, 0, 70, 0, 123, 1, 0, 195, 5, 0, 285, 15, 0, 420, 25, 0, 586, 39, 2, 0, 818, 53, 4, 0, 1110, 73, 6, 0, 1451, 103, 10, 0, 1846, 142, 18, 0, 2361, 181, 26, 0, 2956, 234, 33, 2, 0, 3704, 287, 40, 4, 0, 4567, 348, 49, 8

Offset: 1

David Applegate, Aug 12 2017

Row lengths are A290726(n).
The first entry of each row is 0, because an intersection requires at least 2 lines.
The first row with 3 entries is for n=9, because that is the first configuration with a nontrivial intersection.
Row sums give A290447.

			Triangle begins:
  0;
  0;
  0;
  0,   1;
  0,   5;
  0,  15;
  0,  35;
  0,  70;
  0, 123,   1;
  0, 195,   5;
  0, 285,  15;
  0, 420,  25;
  0, 586,  39,   2;

David Applegate, Table of n, a(n) for n = 1..800
David Applegate, Triangular table T(n,k) for n = 1..100
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.

Cf. A290447, A290726, A290865, A290866.

Sum_{k} T(n,k) * binomial(k,2) = binomial(n,4), because there are binomial(n,4) total pairs of semicircles, and an intersection of k consists of binomial(k,2) of those pairs.

A290865(n) = binomial(n,2) + Sum_{k} T(n,k) * (k-1).

A332723 Irregular table read by rows: Take a line with n equally spaced points with semicircles drawn between them, as in A290447. Then T(n,k) = number of k-sided regions in that figure, where k>=2.

Original entry on oeis.org

1, 2, 1, 3, 4, 4, 10, 0, 1, 5, 19, 3, 3, 6, 31, 13, 6, 7, 46, 35, 10, 8, 65, 74, 14, 9, 92, 131, 18, 10, 140, 192, 27, 1, 11, 202, 274, 46, 3, 12, 275, 396, 62, 3, 13, 363, 563, 79, 9, 14, 467, 784, 100, 14, 15, 598, 1054, 126, 12, 2

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Feb 21 2020

			The first 25 rows are:
1;
2,1;
3,4;
4,10,0,1;
5,19,3,3;
6,31,13,6;
7,46,35,10;
8,65,74,14;
9,92,131,18;
10,140,192,27,1;
11,202,274,46,3;
12,275,396,62,3;
13,363,563,79,9;
14,467,784,100,14;
15,598,1054,126,12,2;
16,772,1358,159,13,2;
17,996,1698,216,24,2,1;
18,1255,2120,266,41,2;
19,1551,2629,346,54,5;
20,1892,3236,425,71,8;
21,2304,3909,525,83,9,1;
22,2793,4676,629,108,9,2;
23,3342,5559,792,125,14,3;
24,3982,6546,948,166,15,2;
25,4705,7658,1145,198,14,3;
The row sums are A290865.

Scott R. Shannon, Table of n, a(n) for n = 2..345. The terms for 2 to 55 points.
Scott R. Shannon, Illustration of regions for n = 5.
Scott R. Shannon, Illustration of regions for n = 6.
Scott R. Shannon, Illustration of regions for n = 10.
Scott R. Shannon, Illustration of regions for n = 11.
Scott R. Shannon, Illustration of regions for n = 15.
Scott R. Shannon, Illustration of regions for n = 16.
Scott R. Shannon, Illustration of regions for n = 20.
Scott R. Shannon, Illustration of regions for n = 21.
Scott R. Shannon, Illustration of regions for n = 25.
Scott R. Shannon, Illustration of regions for n = 26.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 10.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 11.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 15.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 16.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 20.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 21.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 25.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 26.

Cf. A290447, A290865, A290866, A290867, A290726.

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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.

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A290865 a(n) = number of regions in the configuration A290447(n).

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A359254 Number of edges among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

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A290867 Irregular triangle read by rows: the number of points that are the intersections of k semicircles in the configuration A290447(n).

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A332723 Irregular table read by rows: Take a line with n equally spaced points with semicircles drawn between them, as in A290447. Then T(n,k) = number of k-sided regions in that figure, where k>=2.

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