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 1-5 of 5 results.

A384700 On a 2 X n grid of vertices, draw a circle through every unordered triple of non-collinear vertices: a(n) is the number of distinct circles created.

Original entry on oeis.org

0, 1, 9, 24, 52, 93, 153, 232, 336, 465, 625, 816, 1044, 1309, 1617, 1968, 2368, 2817, 3321, 3880, 4500, 5181, 5929
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jun 07 2025

Keywords

Links

Crossrefs

Cf. A384701 (vertices), A384702 (regions), A384703 (edges), A365669, A374338, A373110, A372981,

Formula

Conjecture:
for even n, a(n) = n^3/2 - n^2/4 - n,
for odd n > 1, a(n) = n^3/2 - n^2/4 - n + 3/4.

A384702 On a 2 X n grid of vertices, draw a circle through every unordered triple of non-collinear vertices: a(n) is the number of distinct (finite) regions created.

Original entry on oeis.org

0, 1, 37, 245, 1205, 4213, 12261, 29742, 65507, 130824, 245325, 432262, 727259
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jun 07 2025

Keywords

Comments

The infinite exterior region is not counted.

Links

Crossrefs

Cf. A384700 (circles), A384701 (vertices), A384703 (edges), A359570, A374826, A374337, A372978.

Formula

a(n) = A384703(n) - A384701(n) + 1 by Euler's formula, for n > 1.

A384701 On a 2 X n grid of vertices, draw a circle through every unordered triple of non-collinear vertices: a(n) is the number of distinct points where circles intersect.

Original entry on oeis.org

2, 4, 18, 172, 978, 3672, 11034, 27241, 60804, 122741, 232138, 412263, 697058
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jun 07 2025

Keywords

Links

Crossrefs

Cf. A384700 (circles), A384702 (regions), A384703 (edges), A359569, A374825, A374338, A373106.

Formula

a(n) = A384703(n) - A384702(n) + 1 by Euler's formula, for n > 1.

A385160 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join a circle through every unordered triple of non-collinear points: a(n) is the number of distinct points where circles intersect.

Original entry on oeis.org

4, 68, 4244, 38100, 222300, 695544, 2252764
Offset: 1

Views

Author

Scott R. Shannon, Jul 10 2025

Keywords

Links

  • Scott R. Shannon, Image for n = 2. The 4 x 2 = 8 starting points are shown as white dots.
  • Scott R. Shannon, Image for n = 3. The 4 x 3 = 12 starting points are shown as white dots.

Crossrefs

Cf. A385159 (total circles), A385161 (regions), A385162 (edges), A384703, A383461, A374825, A359569.

Formula

a(n) = A385162(n) - A385161(n) + 1 by Euler's formula.

A385162 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join a circle through every unordered triple of non-collinear points: a(n) is the number of distinct (curved) edges formed from the intersections of the circles.

Original entry on oeis.org

4, 184, 8956, 79272, 455664, 1420624, 4576632
Offset: 1

Views

Author

Scott R. Shannon, Jul 10 2025

Keywords

Links

  • Scott R. Shannon, Image for n = 2. The 4 x 2 = 8 starting points are shown as white dots.

Crossrefs

Cf. A385159 (total circles), A385160 (vertices), A385161 (regions), A384703, A374827, A373108, A359571.

Formula

a(n) = A385160(n) + A385161(n) - 1 by Euler's formula.
Showing 1-5 of 5 results.

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

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