A285521 - OEIS

%I #47 Jan 02 2024 12:45:32

%S 1,1,1,1,1,2,1,1,2,2,1,1,3,2,3,1,2,1,3,2,3,1,3,1,4,4,2,3,2,3,1,1,4,4,

%T 2,3,1,3,4,1,5,5,2,4,3,3,2,4,1,1,5,5,2,4,3,2,1,5,6,1,7,7,2,6,3,4,4,3,

%U 6,2,7,7,1,1,6,2,5,4,2,4,1,7,8,1,9,9,2

%N Table read by rows: the n-th row gives the lexicographically earliest sequence of length n such that the convex hull of (1, a(1)), ..., (n, a(n)) is an n-gon with minimum height.

%C "Minimum height" means that there is no other sequence such that the convex hull is an n-gon and max(a'(1),...,a'(n)) - min(a'(1),...,a'(n)) is smaller.

%C Each number appears at most twice in any given row.

%C Conjecture: Even-length rows have rotational symmetry.

%C Conjecture: The maximum value in any even-length row is the same as the maximum value in the preceding row.

%C Conjecture by Peter Kagey: 24th row is 12, 14, 8, 17, 5, 4, 21, 22, 2, 23, 1, 1, 24, 24, 2, 23, 3, 4, 21, 20, 8, 17, 11, 13. - _Lars Blomberg_, May 06 2017

%H Peter Kagey and Lars Blomberg, <a href="/A285521/b285521.txt">Table of n, a(n) for n = 1..276</a> (23 rows, first 21 rows from Peter Kagey)

%H Mathematics Stack Exchange user Smylic, <a href="https://math.stackexchange.com/a/2244868/121988">The minimum "height" of a convex polygon on N^2</a>.

%e Row 7 is [1,3,1,4,4,2,3], the lexicographically earliest sequence with the minimal height of 3 and a convex hull that forms a heptagon.

%e      xx

%e    x    x

%e        x

%e   x x

%e Row 8 is [2,3,1,1,4,4,2,3], the lexicographically earliest sequence with the minimal height of 3 and a convex hull that forms an octagon.

%e       xx

%e    x     x

%e   x     x

%e     xx

%e Table begins:

%e   1;

%e   1, 1;

%e   1, 1, 2;

%e   1, 1, 2, 2;

%e   1, 1, 3, 2, 3;

%e   1, 2, 1, 3, 2, 3;

%e   1, 3, 1, 4, 4, 2, 3;

%e   2, 3, 1, 1, 4, 4, 2, 3;

%K nonn,tabl

%O 1,6

%A _Peter Kagey_, Apr 20 2017