A371373 -id:A371373 - cp's OEIS frontend

A371374 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of regions formed.

1, 1, 9, 9, 51, 48, 211, 217, 612, 651, 1475, 1248, 3017, 3193, 5415, 5793, 9623, 9000, 15429, 15901, 23352, 24311, 34501, 33840, 49001, 50337, 67365, 69385, 91003, 87720, 120219, 123169, 155430, 159291, 198521, 198792, 250121, 256121, 310635, 317441, 382203, 382032, 465691, 473573

Offset: 1

Scott R. Shannon, Mar 20 2024

nonn

See A371373 and A371254 for further information. The details of the number of regions with k sides is given in A371376.

Scott R. Shannon, Image for n = 3. In this and other images the initial circle and n points are shown in white for clarity.
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 = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 15.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 24.
Scott R. Shannon, Image for n = 30.

Cf. A371373 (vertices), A371375 (edges), A371376 (k-gons), A371377 (vertex crossings), A371254, A371253, A006533, A358782, A359046.

a(n) = A371375(n) - A371373(n) + 1 by Euler's formula.

A371254 Number of vertices formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

Original entry on oeis.org

1, 2, 4, 4, 15, 7, 70, 64, 208, 220, 550, 397, 1131, 1162, 1981, 2128, 3723, 3259, 5966, 6000, 9010, 9240, 13524, 12745, 19325, 19266, 26434, 26684, 35931, 33301, 47368, 47616, 61216, 61676, 78330, 76789, 98901, 99674, 122656, 123560

Offset: 1

Scott R. Shannon, Mar 16 2024

nonn

Other than for n = 3, 4, and 6, all graphs so far investigated in this sequence contain some internal vertices which are created from the intersections of both 2 and 3 arcs, i.e., no graph contains only simple intersections. This is in contrast to the case where the point pairs are connected by straight lines, see A007569 and A335102, where the odd-n graphs contain only simple intersections. See the attached images.
Other patterns for the intersection arc counts are also seen. If n is divisible by 3 then a central vertex is always present that is created from the crossing of n arcs. If n is divisible by 6, then internal vertices are present that are created from the crossing of 6 arcs. For n = 15 and n = 45, internal vertices are present that are created from the crossing of 5 arcs - it is likely all graphs with n = 15+30*k, k>=0, contain such vertices.
For n = 30, the graph also contains internal vertices that are created from the crossing of 9 arcs. It is likely that all graphs with n divisible by 30 contain such vertices. As the graphs created from the straight line diagonal intersections of the regular n-gon, see A007569, also have the maximum possible line intersection count of 7 when n is divisible by 30, it is plausible that 9 is the maximum possible arc intersection count for any internal vertex, other than the central vertex when n is divisible by 3.
Assuming these patterns hold for all n, is it possible that there is a general formula for the number of vertices, analogous to that in A007569 for the intersections of chords in a regular n-gon?

B. Poonen and M. Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, arXiv:math/9508209v3 [math.MG], 1995-2006.
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 = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 15. Note the 5 arc intersections shown in green.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 24.
Scott R. Shannon, Image for n = 30. Note the 9 arc intersections shown in violet.

Cf. A371253 (regions), A371255 (edges), A371274 (k-gons), A371264 (vertex crossings), A370980 (number of circles), A371373 (complete circles), A007569, A335102, A358746, A331702, A359252.

a(n) = A371255(n) - A371253(n) + 1 by Euler's formula.

A371375 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of (curved) edges formed.

Original entry on oeis.org

1, 2, 12, 12, 75, 66, 350, 360, 1071, 1150, 2684, 2148, 5603, 5950, 10110, 10928, 18309, 16830, 29564, 30500, 44961, 46882, 66746, 64872, 95125, 97786, 131112, 135156, 177567, 169770, 235042, 240928, 304359, 312086, 389340, 388764, 491175, 503158, 610662, 624280, 752145, 749742, 917276

Offset: 1

Scott R. Shannon, Mar 20 2024

nonn

See A371373 and A371374 for images of the graphs.

Cf. A371373 (vertices), A371374 (regions), A371376 (k-gons), A371377 (vertex crossings), A371255, A135565, A358783, A359047.

a(n) = A371373(n) + A371374(n) - 1 by Euler's formula.

A372614 Number of vertices among all distinct circles that can be constructed from the 3 vertices and the equally spaced 3*n points placed on the sides of an equilateral triangle, using only a compass.

Original entry on oeis.org

6, 87, 481, 1992, 6969, 15409, 35202, 58422, 107677, 159138, 268572, 350860, 557049

Offset: 0

Scott R. Shannon, May 07 2024

A circle is constructed for every pair of the 3 + 3*n points, the first point defines the circle's center while the second the radius distance.

Scott R. Shannon, Image for n = 0. In this and other images the 3 + 3*n vertices forming the triangle are drawn larger for clarity.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.

Cf. A372615 (regions), A372616 (edges), A372617 (k-gons), A372682 (number of circles), A372731, A371373, A354605, A360351.

a(n) = A372616(n) - A372615(n) + 1 by Euler's formula.

A372731 Number of vertices among all distinct circles that can be constructed from the 3 vertices and the equally spaced 3n points placed on the sides of an equilateral triangle when every pair of the 3 + 3n points are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

6, 51, 301, 1272, 3285, 8401, 16050, 30036, 49801, 80916, 120447, 180307, 249108, 350145, 465898, 618213

Offset: 0

Scott R. Shannon, May 12 2024

A circle is constructed for every pair of the 3 + 3*n points, the two points lying at the ends of a diameter of the circle.

Scott R. Shannon, Image for n = 0. In this and other images the 3 + 3*n vertices forming the triangle are drawn larger for clarity.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.

Cf. A372732 (regions), A372733 (edges), A372734 (k-gons), A372735 (number of circles), A372614, A371373, A354605, A360351.

a(n) = A372733(n) - A372732(n) + 1 by Euler's formula.

A370980 If n is even, (n^2-2*n+2)/2, otherwise (n^2-n+2)/2.

Original entry on oeis.org

1, 1, 1, 4, 5, 11, 13, 22, 25, 37, 41, 56, 61, 79, 85, 106, 113, 137, 145, 172, 181, 211, 221, 254, 265, 301, 313, 352, 365, 407, 421, 466, 481, 529, 545, 596, 613, 667, 685, 742, 761, 821, 841, 904, 925, 991, 1013, 1082, 1105, 1177, 1201, 1276, 1301, 1379, 1405, 1486, 1513, 1597, 1625, 1712, 1741, 1831, 1861, 1954

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 23 2024

nonn

Total number of circles in A371373 and A371253, if in the later all the circular arcs are completed to form full circles.

The sequence also gives the number of vertices created from circle intersections when a circle of radius r is drawn around each of n equally spaced points on the circumference of a circle of radius r. The number of regions in these constructions is A093005(n) and the number of edges is A183207(n). See the attached images. - Scott R. Shannon, Jul 06 2024.

			a(n) = 1+n*floor((n-1)/2) = 1+n*A004526(n-1). - _Chai Wah Wu_, Mar 23 2024

Scott R. Shannon, Image for n = 3. In this and other images the center of each circle of shown as a white dot.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 20.

Bisections are A001844 and A084849.
Cf. A371373, A371374, A371375.
Cf. A004526, A371253, A371254, A371253, A183297, A093005, A183207.

def A370980(n): return n*(n-1>>1)+1 # Chai Wah Wu, Mar 23 2024

a(n) = A183207(n) - A093005(n) + 1, by Euler's formula. - Scott R. Shannon, Jul 07 2024

A371376 Irregular table read by rows: place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. T(n,k), k>=2, gives the number of k-sided regions formed.

Original entry on oeis.org

1, 6, 3, 8, 0, 1, 15, 30, 5, 1, 18, 30, 14, 147, 35, 7, 7, 1, 8, 152, 48, 8, 0, 0, 1, 27, 351, 171, 36, 27, 10, 390, 200, 10, 40, 0, 0, 0, 1, 22, 693, 649, 33, 77, 0, 0, 0, 0, 1, 12, 780, 408, 0, 48, 26, 1404, 1183, 234, 169, 0, 0, 0, 0, 0, 0, 1, 14, 1498, 1274, 224, 154, 14, 14, 0, 0, 0, 0, 0, 1

Offset: 2

Scott R. Shannon, Mar 20 2024

See A371373 and A371374 for images of the graphs.

			The table begins:
1;
6, 3;
8, 0, 1;
15, 30, 5, 1;
18, 30;
14, 147, 35, 7, 7, 1;
8, 152, 48, 8, 0, 0, 1;
27, 351, 171, 36, 27;
10, 390, 200, 10, 40, 0, 0, 0, 1;
22, 693, 649, 33, 77, 0, 0, 0, 0, 1;
12, 780, 408, 0, 48;
26, 1404, 1183, 234, 169, 0, 0, 0, 0, 0, 0, 1;
14, 1498, 1274, 224, 154, 14, 14, 0, 0, 0, 0, 0, 1;
45, 2310, 2400, 390, 255, 15;
16, 2736, 2032, 656, 320, 0, 32, 0, 0, 0, 0, 0, 0, 0, 1;
34, 3978, 4097, 969, 493, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1;
18, 4410, 3078, 972, 468, 36, 18;
76, 6365, 6365, 1596, 855, 95, 76, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
20, 6840, 6000, 2100, 780, 60, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
63, 8946, 10395, 2751, 924, 126, 147;
22, 10076, 9218, 3674, 1166, 22, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
46, 13156, 14996, 4347, 1702, 92, 138, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                                                                     \\ 0, 0, 1;
24, 14232, 13296, 4512, 1440, 96, 240;
100, 19075, 19850, 6975, 2675, 150, 175, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                                                                  \\ 0, 0, 0, 1;
.
.

Cf. A371373 (vertices), A371374 (regions), A371375 (edges), A371377 (vertex crossings), A371274, A331450, A359009, A359061.

Sum of row(n) = A371374(n).

A372977 Number of vertices among all distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square, using only a compass.

Original entry on oeis.org

40, 553, 4204, 14505, 39004, 94365, 197464, 320925, 569600

Offset: 0

Scott R. Shannon, May 19 2024

A circle is constructed for every pair of the 4 + 4*n points, the first point defines the circle's center while the second the radius distance.

Scott R. Shannon, Image for n = 0. In this and other images the 4 + 4*n vertices forming the square are drawn larger for clarity.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.

Cf. A372978 (regions), A372979 (edges), A372980 (k-gons), A372981 (circles), A372614, A372731, A371373, A354605, A360351.

a(n) = A372979(n) - A372978(n) + 1 by Euler's formula.

A374338 Start with two vertices and draw a circle around each whose radius is the distance between the vertices. The sequence gives the number of vertices constructed after n iterations of drawing circles with this same radius around every new vertex created from all circles' intersections. See the Comments.

Original entry on oeis.org

4, 8, 14, 24, 34, 46, 62, 78, 96, 118, 140, 164, 192, 220, 250, 284, 318, 354, 394, 434, 476, 522, 568, 616, 668, 720, 774, 832, 890, 950, 1014, 1078, 1144, 1214, 1284, 1356, 1432, 1508, 1586, 1668, 1750, 1834, 1922, 2010, 2100, 2194, 2288, 2384, 2484, 2584, 2686, 2792

Offset: 1

Scott R. Shannon, Jul 05 2024

nonn

Start with two vertices and, using each as the center, draw a circle around each whose radius is the distance between the vertices. These circles' intersections create two additional vertices, so after the first iteration four vertices exist. Using these four vertices as centers draw four new circles whose radius is the same as the distance between the initial two vertices. These circles' intersections create eight new vertices. Repeat this process n times; the sequence gives the number of vertices after n iterations.

Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 16.

Cf. A374337 (regions), A374339 (edges), A359569, A371373, A371254.

a(n) = A374339(n) - A374337(n) + 1, by Euler's formula.
Conjectured:
If n = 3*k + 1, k >= 0, a(n) = (3*n^2 + 5*n + 4)/3.
If n = 3*k, k >= 1, a(n) = (3*n^2 + 5*n)/3.
If n = 3*k - 1, k >= 1, a(n) = (3*n^2 + 5*n + 2)/3.

A371377 Irregular table read by rows: place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. T(n,k), k>=2, gives the number of vertices formed by the crossing of k arcs.

Original entry on oeis.org

0, 0, 0, 4, 0, 4, 10, 10, 0, 5, 6, 6, 0, 6, 1, 98, 35, 0, 0, 0, 7, 104, 32, 0, 0, 0, 8, 369, 81, 0, 0, 0, 0, 0, 10, 410, 80, 0, 0, 0, 0, 0, 10, 1034, 165, 0, 0, 0, 0, 0, 0, 0, 11, 768, 84, 0, 0, 36, 0, 0, 0, 0, 12, 1, 2288, 286, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 2464, 280, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14

Offset: 1

Scott R. Shannon, Mar 20 2024

See A371373 and A371374 for images of the graphs.

			The table begins:
0;
0;
0, 4;
0, 4;
10, 10, 0, 5;
6, 6, 0, 6, 1;
98, 35, 0, 0, 0, 7;
104, 32, 0, 0, 0, 8;
369, 81, 0, 0, 0, 0, 0, 10;
410, 80, 0, 0, 0, 0, 0, 10;
1034, 165, 0, 0, 0, 0, 0, 0, 0, 11;
768, 84, 0, 0, 36, 0, 0, 0, 0, 12, 1;
2288, 286, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
2464, 280, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14;
4230, 420, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16;
4672, 448, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16;
7990, 680, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17;
7254, 450, 0, 0, 108, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 1;
13148, 969, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19;
13620, 960, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20;
20265, 1323, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22;
21230, 1320, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22;
30452, 1771, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 23;
29376, 1416, 0, 0, 216, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 1;
43800, 2300, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25;
45136, 2288, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26;
.
.

Cf. A371373 (vertices), A371374 (regions), A371375 (edges), A371376 (k-gons), A371264, A335102, A007569, A358746, A331702.

Sum of row(n) = A371373(n).

cp's OEIS Frontend

A371374 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of regions formed.

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A371254 Number of vertices formed when n equally spaced points are placed around a circle and all pairs of points are joined by an interior arc whose radius equals the circle's radius.

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A371375 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of (curved) edges formed.

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A372614 Number of vertices among all distinct circles that can be constructed from the 3 vertices and the equally spaced 3*n points placed on the sides of an equilateral triangle, using only a compass.

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A370980 If n is even, (n^2-2*n+2)/2, otherwise (n^2-n+2)/2.

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A372977 Number of vertices among all distinct circles that can be constructed from the 4 vertices and the equally spaced 4*n points placed on the sides of a square, using only a compass.

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