A357058 - OEIS

%I #22 Sep 17 2022 14:14:24

%S 1,5,17,37,65,93,145,181,257,309,401,457,577,653,785,869,1025,1109,

%T 1297,1413,1601,1725,1937,2041,2305,2453,2705,2861,3137,3289,3601,

%U 3765,4089,4293,4625,4801,5185,5405,5769,5993,6401,6605,7057,7309,7737,8013,8465,8673,9217,9477,9993,10309

%N Number of regions in a square when n internal squares are drawn between the 4n points that divide each side into n+1 equal parts.

%C The even values of n that yield squares with non-simple intersections are 32, 38, 44, 50, 54, 62, 76, 90, 98, ... .

%H Scott R. Shannon, <a href="/A357058/a357058.jpg">Image for n = 1</a>.

%H Scott R. Shannon, <a href="/A357058/a357058_1.jpg">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A357058/a357058_2.jpg">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A357058/a357058_3.jpg">Image for n = 5</a>. This is the first term that forms squares with non-simple intersections.

%H Scott R. Shannon, <a href="/A357058/a357058_4.jpg">Image for n = 10</a>.

%H Scott R. Shannon, <a href="/A357058/a357058_5.jpg">Image for n = 32</a>. This is the first term with n mod 2 = 0 that forms squares with non-simple intersections.

%H Scott R. Shannon, <a href="/A357058/a357058_6.jpg">Image for n = 200</a>.

%F a(n) = A357061(n) - A357060 (n) + 1 by Euler's formula.

%F Conjecture: a(n) = 4*n^2 + 1 for squares that only contain simple intersections when cut by n internal squares. This is never the case for odd n >= 5.

%Y Cf. A357060 (vertices), A357061 (edges), A108914, A355838, A355798, A356984 (triangle).

%K nonn

%O 0,2

%A _Scott R. Shannon_, Sep 10 2022