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Number of vertices among all distinct circles that can be constructed from a 2 x n square grid of points using only a compass.
6

%I #20 Oct 11 2024 09:15:35

%S 2,40,190,740,1824,3956,7314,12956,20684,32276,47348,68516,94550,

%T 128780,170106,222252,283418,358756,445534,550868,670358,811556,

%U 970740,1157168,1363700,1601384,1864524,2164668,2493136,2865176,3269606,3724112,4215536,4762284,5353050

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

%C 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).

%C No formula for a(n) is known.

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A359859%2B60%2B61.py">Python program</a>.

%H Scott R. Shannon, <a href="/A359859/a359859.jpg">Image for n = 1</a>.

%H Scott R. Shannon, <a href="/A359859/a359859_1.jpg">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A359859/a359859_2.jpg">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A359859/a359859_3.jpg">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A359859/a359859_4.jpg">Image for n = 5</a>.

%H Scott R. Shannon, <a href="/A359859/a359859_5.jpg">Image for n = 6</a>.

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

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

%K nonn

%O 1,1

%A _Scott R. Shannon_, Jan 16 2023

%E a(19)-a(35) from _Lucas A. Brown_, Oct 11 2024