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.

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

Author

Scott R. Shannon, Jan 16 2023

Keywords

• nonn

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

• Lucas A. Brown, Python program.
• 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.
• Scott R. Shannon, Image for n = 5.
• Scott R. Shannon, Image for n = 6.

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