cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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.

Original entry on oeis.org

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

Views

Author

Scott R. Shannon, Mar 20 2024

Keywords

Comments

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

Links

Crossrefs

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

Formula

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

A331702 Number of distinct intersections among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

Original entry on oeis.org

0, 2, 6, 40, 55, 145, 238, 584, 612, 1350, 1804, 2401, 3523, 5180, 6150, 9312, 11101, 13645, 17746, 22300, 25998, 33462, 39514, 43993, 55225, 66976, 74088, 88956, 102109, 111841, 133672, 155808, 170940, 198798, 220150, 243937, 275983, 313728, 338208, 382480, 419143, 448561, 507658
Offset: 1

Views

Author

Matej Veselovac, Jan 25 2020

Keywords

Comments

Sequence counts intersections among all distinct circles such that: A circle is defined by a pair of distinct points of a regular n-sided polygon. First point is the center of the circle, while the distance between the points defines the radius of the circle.
It seems one additional intersection exists at the center of the polygon if and only if n is a multiple of 6. From this and n symmetries of the n-sided regular polygon, it would follow that n divides either a(n) or a(n)-1, depending on whether n is a multiple of 6.
A093353(n-1) gives the number of unique circles whose intersections a(n) counts.
From Scott R. Shannon, Dec 15 2022 (Start)
The values for n which lead to all vertices, other than those defining the n-sided regular polygon, being simple start 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, ... . These are all prime values except for the prime squares 4 and 25 which also appear. It is likely all primes appear although what other values lead to only simple vertices is unknown. (End)

Examples

			a(1)=0, we need at least two points to define a radius and a center.
a(2)=2, 2 circles constructed on segment endpoints intersect at 2 points.
a(3)=6, 3 circles on vertices of a triangle intersect at 6 distinct points.
a(4)=40, 8 circles can be constructed on vertices of a square and intersect at 40 distinct points.
a(5)=55, 10 circles can be constructed on vertices of a pentagon and intersect at 55 distinct points.

Links

Crossrefs

Cf. A093353, A359046 (regions), A359047 (edges), A359061 (k-gons), A358746.

Programs

  • GeoGebra
    n = Slider(2, 10, 1);
C = Unique(RemoveUndefined(Flatten(Sequence(Sequence(Circle(Point({cos((2v Pi) / n), sin((2v Pi) / n)}), 2sin((c Pi) / n)), c, 1, floor(n / 2)), v, 1, n))));
I = Unique(RemoveUndefined(Flatten(Sequence(Sequence(Intersect(Element(C, i), Element(C, j)), j, 1, Length(C)), i, 1, Length(C)))));
a_n = Length(I);

Extensions

a(24)-a(30) from Giovanni Resta, Mar 27 2020
a(31)-a(43) from Scott R. Shannon, Dec 14 2022

A353782 Number of regions among all distinct circles that can be constructed from a point on the origin and n equally spaced points on each of the +x,-x,+y,-y coordinates axes using only a compass.

Original entry on oeis.org

112, 1264, 5548, 14976, 37092, 77096, 143560, 237504
Offset: 1

Views

Author

Scott R. Shannon, Mar 13 2023

Keywords

Comments

A circle is constructed for every pair of the 1 + 4n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the points is A361622(n).
No formula for a(n) is currently known.

Links

Crossrefs

Cf. A354605 (vertices), A356358 (edges), A361623 (k-gons), A361622 (distinct circles), A359933, A359860, A359253, A359570, A359046.

Formula

a(n) = A356358 - A354605(n) + 1 by Euler's formula.

A359061 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

Original entry on oeis.org

3, 0, 7, 0, 16, 29, 0, 30, 35, 1, 0, 90, 96, 0, 105, 126, 35, 1, 0, 272, 304, 48, 32, 0, 1, 0, 315, 324, 81, 0, 0, 0, 1, 0, 460, 940, 60, 40, 0, 0, 0, 1, 0, 671, 858, 264, 88, 11, 0, 0, 0, 1, 0, 960, 1656, 108, 48, 0, 1144, 1807, 559, 130, 13, 0, 0, 0, 0, 0, 1, 0, 1960, 3136, 448, 168, 0, 14, 0, 0, 0, 0, 0, 1
Offset: 2

Views

Author

Scott R. Shannon, Dec 14 2022

Keywords

Comments

See A331702 and A359046 for further details and images.
Conjecture: the only value for n which leads to the creation of 2-gons is n = 2. Despite values for n mod 6 = 0 forming intersecting arcs at the center of the n-gon, these are cut by other circles and thus create 3-gons or 4-gons. This is in contrast to values of n mod 4 = 0 in A359009 which do lead to the creation of 2-gons at the center of the figure from similar arcs.

Examples

			The table begins:
3;
0, 7;
0, 16, 29;
0, 30, 35, 1;
0, 90, 96;
0, 105, 126, 35, 1;
0, 272, 304, 48, 32, 0, 1;
0, 315, 324, 81, 0, 0, 0, 1;
0, 460, 940, 60, 40, 0, 0, 0, 1;
0, 671, 858, 264, 88, 11, 0, 0, 0, 1;
0, 960, 1656, 108, 48;
0, 1144, 1807, 559, 130, 13, 0, 0, 0, 0, 0, 1;
0, 1960, 3136, 448, 168, 0, 14, 0, 0, 0, 0, 0, 1;
0, 2100, 3270, 945, 180, 15, 0, 0, 0, 0, 0, 0, 0, 1;
0, 3088, 5584, 896, 368, 16, 16, 0, 0, 0, 0, 0, 0, 0, 1;
0, 3400, 5814, 1513, 493, 85, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 4536, 8712, 1224, 288, 54, 36;
0, 5586, 8797, 2774, 665, 76, 152, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 7940, 12480, 2440, 960, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 7833, 14175, 3486, 1050, 147, 63, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 10428, 19448, 3850, 1408, 22, 44, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

Links

Crossrefs

Cf. A331702 (vertices), A359046 (regions), A359047 (edges), A359009, A358782, A007678.

Formula

Sum of row n = A359046(n).

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

Views

Author

Scott R. Shannon, Dec 22 2022

Keywords

Comments

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.

Links

Crossrefs

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

Formula

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

A359570 Number of regions after n iterations of constructing circles from all current vertices using only a compass, starting with one vertex. See the Comments.

Original entry on oeis.org

0, 1, 3, 21, 7169
Offset: 1

Views

Author

Scott R. Shannon, Jan 06 2023

Keywords

Comments

See A359569 for further details and images.

Links

Crossrefs

Cf. A359569 (vertices), A359571 (edges), A359619 (k-gons), A359253, A359046, A358782.

Formula

For n >= 3, a(n) = A359571(n) - A359569(n) + 1 by Euler's formula.

A359047 Number of distinct edges among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

Original entry on oeis.org

1, 4, 12, 84, 120, 330, 504, 1240, 1332, 2850, 3696, 5172, 7176, 10906, 12660, 19280, 22440, 28494, 35796, 46220, 52752, 68662, 79488, 91272, 111000, 136838, 149472, 181972, 204972, 229650, 268212, 317024, 343860, 404090, 441420, 496764, 553224, 636538, 679224, 776200, 839844, 914634, 1017036
Offset: 1

Views

Author

Scott R. Shannon, Dec 14 2022

Keywords

Comments

See A331702 and A359046 for further details and images.
No formula for a(n) is currently known.

Crossrefs

Cf. A331702 (vertices), A359046 (regions), A359061 (k-gons), A358783, A135565.

Formula

a(n) = A331702(n) + A359046(n) - 1 by Euler's formula.

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

Views

Author

Scott R. Shannon, Mar 16 2024

Keywords

Comments

See A371254 for further information.

Links

Crossrefs

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

Formula

a(n) = A371255(n) - A371254(n) + 1 by Euler's formula.
Showing 1-8 of 8 results.

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