A290865 -id:A290865 - cp's OEIS frontend

A290876 A290865(n) -(n-1).

0, 0, 1, 4, 11, 25, 50, 91, 153, 241, 360, 525, 736, 1014, 1365, 1792, 2304, 2937, 3684, 4585, 5632, 6831, 8217, 9835, 11659, 13723, 16060, 18673, 21576, 24858, 28441, 32460, 36875, 41718, 47046, 52841, 59069, 65928, 73402, 81482, 90136

Offset: 1

N. J. A. Sloane, Aug 13 2017

nonn

This is the number of regions in the configuration A290447(n) if we remove the baseline.

Cf. A290447, A290865.

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.

Original entry on oeis.org

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

A359253 Number of regions 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

3, 14, 51, 116, 255, 466, 821, 1296, 2003, 2904, 4171, 5726, 7795, 10266, 13399, 17026, 21537, 26702, 32995, 40110, 48511, 57996, 69121, 81376, 95511, 111130, 128953, 148432, 170595

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).

No formula for a(n) is currently known.

Scott R. Shannon, Image for n = 2. In this and other images the n points are shown as white dots.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 20.

Cf. A359252 (vertices), A359254 (edges), A359258 (k-gons), A001859, A290865, A359046, A358782.

a(n) = A359254(n) - A359252(n) + 1 by Euler's formula.

A290866 a(n) = number of segments (edges) in the configuration A290447(n).

Original entry on oeis.org

0, 1, 3, 8, 20, 45, 91, 168, 285, 450, 670, 981, 1375, 1902, 2568, 3371, 4326, 5522, 6927, 8639, 10624, 12882, 15489, 18559, 22006, 25904, 30321, 35254, 40728, 46959, 53721, 61354, 69734, 78917, 89029, 100018, 111758, 124759, 138943

Offset: 1

David Applegate, Aug 12 2017

nonn

Only edges above the line are counted. Total edges = a(n) + n - 1.

David Applegate, Table of n, a(n) for n = 1..100

Cf. A290447, A290865, A290867.

a(n) = A290447(n) + A290865(n).

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.

A333519 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

Original entry on oeis.org

0, 2, 13, 48, 141, 312, 652, 1160, 1978, 3106, 4775, 6826, 9803, 13328, 17904, 23536, 30652, 38640, 48945, 60300, 74248, 89892, 108768, 128990, 153826, 180206, 211483, 245000, 284375, 325140, 374450, 425312, 484168, 545938, 616981, 690132, 775077, 862220

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 26 2020

nonn

A semicircular polygon with 2n+2 points is created by placing n+2 equally spaced vertices along the semicircle's arc (including the two end vertices). Also place n+2 equally spaced vertices along the diameter (again including the same two end vertices). Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Lars Blomberg, Table of n, a(n) for n = 0..50
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 20.

Cf. A007678, A290865, A255011, A332953, A333282, A334458, A334459.

a(21) and beyond from Lars Blomberg, May 01 2020

A333642 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and three equally spaced points along the diameter (a total of n+3 points). See Comments for precise definition.

Original entry on oeis.org

2, 8, 20, 43, 80, 139, 224, 324, 510, 730, 992, 1373, 1820, 2187, 3040, 3844, 4720, 5916, 7220, 8498, 10472, 12463, 14570, 17278, 20150, 23130, 26964, 30961, 34688, 40265, 45632, 51138, 57970, 65008, 72322, 80979, 89984, 99197, 110240, 121570, 132896, 146818

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 31 2020

nonn

A semicircular polygon with n+3 points is created by placing n+2 equally spaced vertices along the semicircle's arc (including the two end vertices). Also place three equally spaced vertices along the diameter; these are the same two end vertices plus one dividing the diameter. Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Lars Blomberg, Table of n, a(n) for n = 1..100
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 7.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 12.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 17.
Scott R. Shannon, Illustration for n = 19.
Scott R. Shannon, Illustration for n = 20.
Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 19 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 20 with random distance-based coloring.
Wikipedia, Semicircle.

Cf. A330914 (n-gons), A330911 (edges), A330913 (vertices), A333643, A333519, A007678, A290865, A092867, A331452, A331929, A331931.

a(21) and beyond from Lars Blomberg, May 03 2020

A333643 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around the arc of a semicircle. See Comments for precise definition.

Original entry on oeis.org

1, 4, 11, 25, 50, 91, 154, 234, 375, 550, 769, 1079, 1456, 1783, 2500, 3196, 3987, 5016, 6175, 7348, 9086, 10879, 12836, 15250, 17875, 20682, 24129, 27811, 31419, 36425, 41416, 46664, 52921, 59500, 66489, 74481, 82954, 91807, 102050, 112750, 123700, 136654

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 31 2020

nonn

A semicircular polygon with n+2 points is created by placing n+2 equally spaced vertices along a semicircle's arc, which includes the two end vertices. Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Note that there is a curious relationship between the terms of this sequence and the number of regions in the 'general position' polygon given in A006522. They are a match except for every third term starting at a(8) = 234. Examining the images for n = 8,11,14,17 shows that these polygons have interior points at which three or more lines intersect, while the other n values have no such intersection points. Such multi-line intersection points will reduce the number of regions as compared to the general position polygon which has no multi-line intersection points. This is reflected by the terms in this sequence being lower than the corresponding value in A006522 for n = 8,11,14,... . Why every third value of n in this sequence starting at n = 8 leads to polygons having multiple line intersection points while other values of n do not is currently not known.

Lars Blomberg, Table of n, a(n) for n = 1..80
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 7.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 12.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 17.
Scott R. Shannon, Illustration for n = 19.
Scott R. Shannon, Illustration for n = 20.
Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 19 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 20 with random distance-based coloring.
Wikipedia, Semicircle.

Cf. A333642, A333519, A007678, A290865, A092867, A331452, A331929, A331931.

More terms from Lars Blomberg, Apr 20 2020

cp's OEIS Frontend

A290876 A290865(n) -(n-1).

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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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A359253 Number of regions among all distinct circles that can be constructed from n equally spaced points along a line using only a compass.

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A290866 a(n) = number of segments (edges) in the configuration A290447(n).

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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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A333519 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

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A333642 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and three equally spaced points along the diameter (a total of n+3 points). See Comments for precise definition.

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A333643 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around the arc of a semicircle. See Comments for precise definition.

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