%I A003003 M0439
%S A003003 1,2,3,3,4,5,5,6,7,8,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,
%T A003003 17,17,18,18,18,19,20,20,20,21,21,21,22,22,22,23,23,24,24,24,25,25,26,
               26
%N A003003 Size of the largest subset of the numbers [1...n] which doesn't contain 
               a 4-term arithmetic progression.
%C A003003 These subsets have been called 4-free sequences.
%C A003003 Szemeredi's theorem for arithmetic progressions of length 4 asserts that 
               a(n) is o(n) as n -> infinity. - Doron Zeilberger, Mar 26 2008
%C A003003 False g.f. (z**12+1-z**11-z**10+z**8-z**6+z**5-z**3+z)/((z+1)*(z-1)**2) 
               was conjectured by S. Plouffe in his 1992 dissertation, but in fact 
               is wrong (cf. A136746).
%D A003003 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003003 S. S. Wagstaff, Jr., On k-free sequences of integers, Math. Comp., 26 
               (1972), 767-771.
%H A003003 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A003003 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%Y A003003 Cf. A003002, A003004, A003005, A065825.
%Y A003003 Sequence in context: A067022 A113818 A136746 this_sequence A049474 A076874 
               A127041
%Y A003003 Adjacent sequences: A003000 A003001 A003002 this_sequence A003004 A003005 
               A003006
%K A003003 nonn,more
%O A003003 1,2
%A A003003 N. J. A. Sloane (njas(AT)research.att.com).