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A338886 a(n) is the number of positive integers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.

%I A338886 #28 Sep 11 2021 19:16:51
%S A338886 0,1,1,2,3,3,5,6,7,9,12,11,15,15,16,19,24,20,28,25,29,30,36,33,44,40,
%T A338886 42,41,51,44,59,52,55,57,69,56,76,68,71,73,89,72,92,81,89,90,107,86,
%U A338886 115,101,107,101,129,103,126,117,122,126,147,113,153,136,148
%N A338886 a(n) is the number of positive integers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.
%C A338886 A diagonal lattice rectangle is a rectangle with integer coordinates and no side parallel to the x-axis.
%C A338886 This sequence gives the row lengths of A338885.
%H A338886 Peter Kagey, <a href="/A338886/b338886.txt">Table of n, a(n) for n = 1..1000</a>
%H A338886 Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/q/213754/53884">Rectangles in rectangles</a>
%F A338886 a(n) >= A338887(n).
%e A338886 For n = 5 there are a(5) = 3 different y-values that appear in the coordinates of diagonal lattice rectangles that touch the x-axis, the y-axis, and the line x = 5. An example of each, listed by vertices counterclockwise:
%e A338886    y_max = 4: (4,4), (0,2), (1,0), (5,2);
%e A338886    y_max = 5: (4,5), (0,4), (1,0), (5,1);
%e A338886    y_max = 7: (3,7), (0,6), (2,0), (5,1).
%Y A338886 Cf. A085582, A113751, A338885, A338887.
%K A338886 nonn
%O A338886 1,4
%A A338886 _Peter Kagey_, Nov 14 2020