cp's OEIS Frontend

A357058 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.

%I A357058 #22 Sep 17 2022 14:14:24
%S A357058 1,5,17,37,65,93,145,181,257,309,401,457,577,653,785,869,1025,1109,
%T A357058 1297,1413,1601,1725,1937,2041,2305,2453,2705,2861,3137,3289,3601,
%U A357058 3765,4089,4293,4625,4801,5185,5405,5769,5993,6401,6605,7057,7309,7737,8013,8465,8673,9217,9477,9993,10309
%N A357058 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 A357058 The even values of n that yield squares with non-simple intersections are 32, 38, 44, 50, 54, 62, 76, 90, 98, ... .
%H A357058 Scott R. Shannon, <a href="/A357058/a357058.jpg">Image for n = 1</a>.
%H A357058 Scott R. Shannon, <a href="/A357058/a357058_1.jpg">Image for n = 2</a>.
%H A357058 Scott R. Shannon, <a href="/A357058/a357058_2.jpg">Image for n = 3</a>.
%H A357058 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 A357058 Scott R. Shannon, <a href="/A357058/a357058_4.jpg">Image for n = 10</a>.
%H A357058 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 A357058 Scott R. Shannon, <a href="/A357058/a357058_6.jpg">Image for n = 200</a>.
%F A357058 a(n) = A357061(n) - A357060 (n) + 1 by Euler's formula.
%F A357058 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 A357058 Cf. A357060 (vertices), A357061 (edges), A108914, A355838, A355798, A356984 (triangle).
%K A357058 nonn
%O A357058 0,2
%A A357058 _Scott R. Shannon_, Sep 10 2022