A371374 -id:A371374 - cp's OEIS frontend

A371373 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 vertices formed.

1, 2, 4, 4, 25, 19, 140, 144, 460, 500, 1210, 901, 2587, 2758, 4696, 5136, 8687, 7831, 14136, 14600, 21610, 22572, 32246, 31033, 46125, 47450, 63748, 65772, 86565, 82051, 114824, 117760, 148930, 152796, 190820, 189973, 241055, 247038, 300028, 306840, 369943, 367711, 451586, 459448

Offset: 1

Scott R. Shannon, Mar 20 2024

nonn

The number of circles that cross to form the intersections follows a similar pattern to that seen in A371254; see that sequence for further information. The details of the crossing counts are given in A371377.

Scott R. Shannon, Image for n = 3. In this and other images the initial circle on which the n points are placed is drawn in a thicker white line 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. Note the 6 circle intersections shown in blue.
Scott R. Shannon, Image for n = 15. Note the 5 circle 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 circle intersections shown in violet.

Cf. A371374 (regions), A371375 (edges), A371376 (k-gons), A371377 (vertex crossings), A371254, A007569, A358746, A331702.

a(n) = A371375(n) - A371374(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.

A372615 Number of regions 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

7, 121, 624, 2368, 7912, 17115, 38461, 63184, 115614, 170149, 285715, 371982, 588043

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.

The vertices of the initial equilateral triangle are indicated by small circles in the illustrations here.

Scott R. Shannon, Image for n = 0.
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. A372614 (vertices), A372616 (edges), A372617 (k-gons), A372682 (number of circles), A371374, A353782, A360352.

a(n) = A372616(n) - A372614(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

A371253 Number of regions 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, 1, 6, 5, 26, 18, 99, 89, 270, 271, 650, 516, 1288, 1303, 2250, 2337, 4047, 3636, 6404, 6401, 9597, 9769, 14261, 13632, 20251, 20125, 27594, 27749, 37324, 35040, 49043, 49185, 63228, 63547, 80676, 79380, 101640, 102259, 125853, 126561

Offset: 1

Scott R. Shannon, Mar 16 2024

nonn

See A371254 for further information.

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 = 16.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 24.

Cf. A371254 (vertices), A371255 (edges), A371274 (k-gons), A370980 (number of circles), A371374 (complete circles), A006533, A358782, A359046, A359253, A007678.

a(n) = A371255(n) - A371254(n) + 1 by Euler's formula.

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).

A372978 Number of regions 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

45, 628, 4633, 15476, 41561, 98808, 206317, 333272, 590181

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 as white dots for clarity.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.

Cf. A372977 (vertices), A372979 (edges), A372980 (k-gons), A372981 (circles), A372615, A371374, A353782, A360352.

a(n) = A372979(n) - A372977(n) + 1 by Euler's formula.

A372732 Number of regions 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

7, 88, 438, 1567, 3910, 9504, 17905, 32833, 54066, 86365, 128380, 190152, 262699, 365761, 486711, 642154

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.
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. A372731 (vertices), A372733 (edges), A372734 (k-gons), A372735 (number of circles), A372615, A371374, A353782, A360352.

a(n) = A372733(n) - A372731(n) + 1 by Euler's formula.

A374337 Start with two vertices and draw a circle around each whose radius is the distance between the vertices. The sequence gives the number of regions constructed after n iterations of drawing circles with this same radius around every new vertex created from all circles' intersections.

Original entry on oeis.org

3, 11, 27, 55, 99, 145, 203, 277, 353, 441, 545, 651, 769, 903, 1039, 1187, 1351, 1517, 1695, 1889, 2085, 2293, 2517, 2743, 2981, 3235, 3491, 3759, 4043, 4329, 4627, 4941, 5257, 5585, 5929, 6275, 6633, 7007, 7383, 7771, 8175, 8581, 8999, 9433, 9869, 10317, 10781, 11247, 11725, 12219, 12715

Offset: 1

Scott R. Shannon, Jul 05 2024

nonn

See A374338 for further details.

Scott R. Shannon, Image for n = 1. In this and other images the initial vertices that form the circles' centers are shown as white dots.
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. A374338 (vertices), A374339 (edges), A359570, A371374, A371253.

a(n) = A374339(n) - A374338(n) + 1, by Euler's formula.
Conjectured:
If n = 3*k + 1, k >= 0, a(n) = |(15*n^2 - 17*n - 7)/3|.
If n = 3*k, k >= 1, a(n) = (15*n^2 - 17*n - 3)/3.
If n = 3*k - 1, k >= 1, a(n) = (15*n^2 - 17*n + 7)/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

A371373 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 vertices formed.

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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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A372615 Number of regions 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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A371253 Number of regions 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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A372978 Number of regions 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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A374337 Start with two vertices and draw a circle around each whose radius is the distance between the vertices. The sequence gives the number of regions constructed after n iterations of drawing circles with this same radius around every new vertex created from all circles' intersections.

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