A338885 - OEIS

%I #22 Sep 11 2021 18:55:12

%S 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 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 15,18,20,21,28,33,36,37,7,8,11,12,13,14,15,17,20,22,23

%N 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 A diagonal lattice rectangle is a rectangle with integer coordinates and no side parallel to the x-axis.

%C Conjecture: The smallest number in the n-th row is A228286(n).

%C Conjecture: The largest number in the n-th row is A033638(n).

%H Peter Kagey, <a href="/A338885/b338885.txt">Table of n, a(n) for n = 2..11808</a> (first 100 rows, flattened)

%H Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/q/213754/53884">Rectangles in rectangles</a>

%e Table begins:

%e    n | n-th row

%e -----+------------------------------------------------

%e    2 | 2

%e    3 | 3

%e    4 | 4,  5

%e    5 | 4,  5,  7

%e    6 | 6,  9, 10

%e    7 | 5,  7,  8, 11, 13

%e    8 | 7,  8, 10, 13, 16, 17

%e    9 | 6,  9, 11, 12, 15, 19, 21

%e   10 | 6,  8, 10, 11, 14, 17, 22, 25, 26

%e   11 | 7,  9, 10, 11, 13, 14, 16, 17, 19, 25, 29, 31

%e   12 | 9, 12, 13, 15, 18, 20, 21, 28, 33, 36, 37

%e For n = 6, three of the diagonal lattice rectangles that touch the y-axis, x-axis, and line x = 6 are:

%e (2 ,6), (0,2), (4,0), (6,4);

%e (2, 9), (0,8), (4,0), (6,1); and

%e (3,10), (0,9), (3,0), (6,1);

%e which have maximum y-values of 6, 9, and 10 respectively.

%Y Cf. A033638, A085582, A113751, A228286.

%Y Cf. A338886 (row lengths).

%K nonn,tabf

%O 2,1

%A _Peter Kagey_, Nov 14 2020