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.

A354605 Number of vertices 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

101, 1145, 5001, 13753, 34497, 72185, 135157, 224321
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. A353782 (regions), A356358 (edges), A361623 (k-gons), A361622 (distinct circles), A359932, A359859, A359252, A359569, A331702.

Formula

a(n) = A356358 - A353782(n) + 1 by Euler's formula.

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.

A359258 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, 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, 0, 8, 4, 2, 0, 22, 23, 4, 2, 0, 50, 52, 12, 2, 0, 110, 103, 36, 6, 0, 190, 200, 64, 12, 0, 314, 387, 88, 28, 4, 0, 498, 606, 152, 32, 8, 0, 770, 941, 228, 58, 4, 2, 0, 1132, 1352, 338, 68, 12, 2, 0, 1602, 1935, 532, 98, 4, 0, 2122, 2798, 684, 106, 16, 0, 2850, 3843, 940, 132, 24, 6
Offset: 2

Views

Author

Scott R. Shannon, Dec 23 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).
See A359252 and A359253 for other images of the circles.

Examples

			The table begins:
  3;
  0,     8,     4,    2;
  0,    22,    23,    4,   2;
  0,    50,    52,   12,   2;
  0,   110,   103,   36,   6;
  0,   190,   200,   64,  12;
  0,   314,   387,   88,  28,  4;
  0,   498,   606,  152,  32,  8;
  0,   770,   941,  228,  58,  4,  2;
  0,  1132,  1352,  338,  68, 12,  2;
  0,  1602,  1935,  532,  98,  4;
  0,  2122,  2798,  684, 106, 16;
  0,  2850,  3843,  940, 132, 24,  6;
  0,  3774,  4998, 1268, 192, 28,  6;
  0,  4950,  6475, 1644, 276, 44, 10;
  0,  6190,  8454, 1978, 326, 74,  4;
  0,  7778, 10737, 2520, 434, 52, 12, 4;
  0,  9674, 13224, 3202, 528, 58, 12, 4;
  0, 11978, 16169, 4116, 640, 68, 20, 4;
  ...

Links

Crossrefs

Cf. A359253 (regions), A359252 (vertices), A359254 (edges), A001859, A332723, A359061, A359009.

Formula

Sum of row n = A359253(n);

A359569 Number of vertices 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

1, 2, 4, 14, 6562
Offset: 1

Views

Author

Scott R. Shannon, Jan 06 2023

Keywords

Comments

Start with one vertex and a compass. Only a single circle can be drawn, using the vertex as its center, on whose circumference one additional vertex can be arbitrarily placed. From this single pair of vertices two circles can now be drawn, each circle's center being one vertex while the other defines its radius distance. These circles' intersections create two additional vertices, so now four vertices exist. Continue using all vertex pairs to draw distinct circles whose intersections create additional vertices for the next iteration. The sequence gives the number of vertices after n iterations of this process.
An estimate for a(6) can be obtained by calculating the number of distinct circles generated from the 6562 vertices of the fifth iteration - this is approximately 42.1 million circles. The 6562 vertices of the fifth iteration are created from 114 circles, implying the number of vertices per circle is about half the number of circles. Assuming this holds for the sixth iteration leads to an estimate for a(6) of about 886*10^12. The exact number is possibly within reach of numerical calculation, although obtaining a(7) would almost certainly require a theoretical approach.

Links

Crossrefs

Cf. A359570 (regions), A359571 (edges), A359619 (k-gons), A365669 (circles), A359252, A331702, A358746.

Formula

For n >= 3, a(n) = A359571(n) - A359570(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

Views

Author

Scott R. Shannon, Mar 16 2024

Keywords

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

Crossrefs

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

Formula

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

A359254 Number of edges 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

4, 26, 96, 216, 480, 882, 1564, 2464, 3804, 5502, 7912, 10864, 14816, 19526, 25508, 32392, 40992, 50818, 62852, 76432, 92460, 110590, 131904, 155306, 182316, 212188, 246316, 283586, 326100
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).
See A359252 and A359253 for images of the circles.

Crossrefs

Cf. A359252 (vertices), A359253 (regions), A359258 (k-gons), A001859, A290866, A359047, A358783.

Formula

a(n) = A359252(n) + A359253(n) - 1 by Euler's formula.

A359932 Number of vertices among all distinct circles that can be constructed from an n x n square grid of points using only a compass.

Original entry on oeis.org

40, 689, 7240, 38729, 151584, 488741
Offset: 2

Views

Author

Scott R. Shannon, Jan 19 2023

Keywords

Comments

A circle is constructed for every pair of the n x n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the n x n points is A359931(n).
No formula for a(n) is known.

Links

Crossrefs

Cf. A359933 (regions), A359934 (edges), A359935 (k-gons), A359931 (distinct circles), A359859, A359252.

Formula

a(n) = A359934(n) - A359933(n) + 1 by Euler's formula.

A359859 Number of vertices among all distinct circles that can be constructed from a 2 x n square grid of points using only a compass.

Original entry on oeis.org

2, 40, 190, 740, 1824, 3956, 7314, 12956, 20684, 32276, 47348, 68516, 94550, 128780, 170106, 222252, 283418, 358756, 445534, 550868, 670358, 811556, 970740, 1157168, 1363700, 1601384, 1864524, 2164668, 2493136, 2865176, 3269606, 3724112, 4215536, 4762284, 5353050
Offset: 1

Views

Author

Scott R. Shannon, Jan 16 2023

Keywords

Comments

A circle is constructed for every pair of the 2 x n points, the first point defines the circle's center while the second the radius distance. The number of distinct circles constructed from the 2 x n points is conjectured to be 4*A001859(n-1).
No formula for a(n) is known.

Links

Crossrefs

Cf. A359860 (regions), A359861 (edges), A359862 (k-gons), A001859, A359252.

Formula

a(n) = A359861(n) - A359860(n) + 1 by Euler's formula.

Extensions

a(19)-a(35) from Lucas A. Brown, Oct 11 2024
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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