A384701 -id:A384701 - cp's OEIS frontend

A384703 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 edges in the planar graph formed from the intersections of the circles.

Original entry on oeis.org

0, 4, 54, 416, 2182, 7884, 23294, 56982, 126310, 253564, 477462, 844524, 1424316

Offset: 1

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

The edges being counted are of course arcs of circles.

Scott R. Shannon, Image for a(3) = 54.

Cf. A384700 (circles), A384701 (vertices), A384702 (regions), A359571, A374827, A374339, A373108.

a(n) = A384701(n) + A384702(n) - 1 by Euler's formula, for n > 1.

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

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

Scott R. Shannon, Image for a(3) = 9.

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

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

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

The infinite exterior region is not counted.

Scott R. Shannon, Image for a(3) = 37.
Scott R. Shannon, Image for a(4) = 245.
Scott R. Shannon, Image for a(5) = 1205.

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

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

cp's OEIS Frontend

A384703 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 edges in the planar graph formed from the intersections of the circles.

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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.

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Author

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Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula