A094064 - OEIS

%I #13 Jan 13 2016 03:12:39

%S 2,1,5,4,3,10,9,8,7,6,17,16,15,14,13,12,11,26,25,24,23,22,21,20,19,18,

%T 37,36,35,34,33,32,31,30,29,28,27,50,49,48,47,46,45,44,43,42,41,40,39,

%U 38,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,82

%N Sequences has the properties shown in the Comments lines.

%C It contains an infinite increasing subsequence.

%C For each k there is a decreasing subsequence of length > k but no infinite decreasing subsequence.

%C For each n the first n^2 + 1 terms contain a decreasing subsequence of length n + 1 but no increasing subsequence of length n + 1.

%H Robert Israel, <a href="/A094064/b094064.txt">Table of n, a(n) for n = 0..10000</a>

%H P. Erdõs, G. Szekeres, <a href="http://www.renyi.hu/~p_erdos/1935-01.pdf">A combinatorial problem in geometry</a>, Compositio Math. 2 (1935), 463-470; Zentralblatt 12,270.

%H Lajos Pinter, <a href="http://www.jstor.org/stable/3621353">On monotone subsequences</a>, Math. Gaz., 88 (#511, 2004), 110-111.

%F G.f. (3 + 5*x^2 + 4*x*Sum_{n>=2} n*x^(n^2))/(1-x) - 1/(1-x)^2. - _Robert Israel_, Jan 13 2016

%p A[0]:= 2:

%p A[1]:= 1:

%p for n from 1 to 9 do

%p    for i from 1 to 2*n+1 do

%p      A[n^2+i]:= (n+1)^2+2-i

%p od od:

%p seq(A[i],i=0..100); # _Robert Israel_, Jan 13 2016

%K nonn

%O 0,1

%A _R. K. Guy_, May 01 2004