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A338885 Irregular triangle read by rows in which the n-th row lists all numbers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.

%I A338885 #22 Sep 11 2021 18:55:12
%S A338885 2,3,4,5,4,5,7,6,9,10,5,7,8,11,13,7,8,10,13,16,17,6,9,11,12,15,19,21,
%T A338885 6,8,10,11,14,17,22,25,26,7,9,10,11,13,14,16,17,19,25,29,31,9,12,13,
%U A338885 15,18,20,21,28,33,36,37,7,8,11,12,13,14,15,17,20,22,23
%N A338885 Irregular triangle read by rows in which the n-th row lists all numbers k such that there exists a diagonal lattice rectangle touching all four sides of an n X k rectangle.
%C A338885 A diagonal lattice rectangle is a rectangle with integer coordinates and no side parallel to the x-axis.
%C A338885 Conjecture: The smallest number in the n-th row is A228286(n).
%C A338885 Conjecture: The largest number in the n-th row is A033638(n).
%H A338885 Peter Kagey, <a href="/A338885/b338885.txt">Table of n, a(n) for n = 2..11808</a> (first 100 rows, flattened)
%H A338885 Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/q/213754/53884">Rectangles in rectangles</a>
%e A338885 Table begins:
%e A338885    n | n-th row
%e A338885 -----+------------------------------------------------
%e A338885    2 | 2
%e A338885    3 | 3
%e A338885    4 | 4,  5
%e A338885    5 | 4,  5,  7
%e A338885    6 | 6,  9, 10
%e A338885    7 | 5,  7,  8, 11, 13
%e A338885    8 | 7,  8, 10, 13, 16, 17
%e A338885    9 | 6,  9, 11, 12, 15, 19, 21
%e A338885   10 | 6,  8, 10, 11, 14, 17, 22, 25, 26
%e A338885   11 | 7,  9, 10, 11, 13, 14, 16, 17, 19, 25, 29, 31
%e A338885   12 | 9, 12, 13, 15, 18, 20, 21, 28, 33, 36, 37
%e A338885 For n = 6, three of the diagonal lattice rectangles that touch the y-axis, x-axis, and line x = 6 are:
%e A338885 (2 ,6), (0,2), (4,0), (6,4);
%e A338885 (2, 9), (0,8), (4,0), (6,1); and
%e A338885 (3,10), (0,9), (3,0), (6,1);
%e A338885 which have maximum y-values of 6, 9, and 10 respectively.
%Y A338885 Cf. A033638, A085582, A113751, A228286.
%Y A338885 Cf. A338886 (row lengths).
%K A338885 nonn,tabf
%O A338885 2,1
%A A338885 _Peter Kagey_, Nov 14 2020