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Place n equally space 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.
+10
11
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
COMMENTS
See A371373 and A371254 for further information. The details of the number of regions with k sides is given in A371376.
LINKS
Scott R. Shannon, Image for n = 3. In this and other images the initial circle and n points are shown in white for clarity.
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.
+10
9
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
COMMENTS
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?
LINKS
Scott R. Shannon, Image for n = 15. Note the 5 arc intersections shown in green. Scott R. Shannon, Image for n = 30. Note the 9 arc intersections shown in violet.
CROSSREFS
Cf. A371253 (regions), A371255 (edges), A371274 (k-gons), A371264 (vertex crossings), A370980 (number of circles), A371373 (complete circles), A007569, A335102, A358746, A331702, A359252.
Place n equally space 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.
+10
9
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
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.
+10
8
6, 87, 481, 1992, 6969, 15409, 35202, 58422, 107677, 159138, 268572, 350860, 557049
COMMENTS
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.
LINKS
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.
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 when every pair of the 3 + 3*n points are connected by a circle and where the points lie at the ends of the circle's diameter.
+10
8
6, 51, 301, 1272, 3285, 8401, 16050, 30036, 49801, 80916, 120447, 180307, 249108, 350145, 465898, 618213
COMMENTS
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.
LINKS
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.
Irregular table read by rows: place n equally space 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.
+10
7
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
EXAMPLE
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;
.
.
If n is even, (n^2-2*n+2)/2, otherwise (n^2-n+2)/2.
+10
6
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
COMMENTS
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.
LINKS
Scott R. Shannon, Image for n = 3. In this and other images the center of each circle of shown as a white dot.
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.
+10
6
40, 553, 4204, 14505, 39004, 94365, 197464, 320925, 569600
COMMENTS
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.
LINKS
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.
Irregular table read by rows: place n equally space 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.
+10
4
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
EXAMPLE
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;
.
.
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.
+10
4
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
COMMENTS
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.
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.
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