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A020655 Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 5.

%I A020655 #19 Jan 04 2016 17:38:01
%S A020655 1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19,26,27,28,29,31,32,33,34,36,
%T A020655 37,38,39,41,42,43,44,51,52,53,54,56,57,58,59,61,62,63,64,66,67,68,69,
%U A020655 76,77,78,79,81,82,83,84,86,87,88,89,91,92,93,94,126,127,128,129,131,132,133
%N A020655 Lexicographically earliest increasing sequence of positive numbers that contains no arithmetic progression of length 5.
%H A020655 Robert Israel, <a href="/A020655/b020655.txt">Table of n, a(n) for n = 1..10000</a>
%p A020655 Noap:= proc(N,m)
%p A020655 # N terms of earliest increasing seq with no m-term arithmetic progression
%p A020655 local A,forbid,n,c,ds,j;
%p A020655 A:= Vector(N):
%p A020655 A[1..m-1]:= <($1..m-1)>:
%p A020655 forbid:= {m}:
%p A020655 for n from m to N do
%p A020655   c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);
%p A020655   A[n]:= c;
%p A020655   ds:= convert(map(t -> c-t, A[m-2..n-1]),set);
%p A020655   for j from m-2 to 2 by -1 do
%p A020655     ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]),set);
%p A020655     if ds = {} then break fi;
%p A020655   od;
%p A020655   forbid:= select(`>`,forbid,c) union map(`+`,ds,c);
%p A020655 od:
%p A020655 convert(A,list)
%p A020655 end proc:
%p A020655 Noap(100, 5); # _Robert Israel_, Jan 04 2016
%t A020655 t = {1, 2, 3, 4}; Do[s = Table[Append[i, n], {i, Subsets[t, {4}]}]; If[! MemberQ[Table[Differences[i, 2], {i, s}], {0, 0, 0}], AppendTo[t, n]], {n, 5, 100}]; t (* _T. D. Noe_, Apr 17 2014 *)
%Y A020655 Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
%Y A020655 3-term AP: A005836 (>=0), A003278 (>0);
%Y A020655 4-term AP: A005839 (>=0), A005837 (>0);
%Y A020655 5-term AP: A020654 (>=0), A020655 (>0);
%Y A020655 6-term AP: A020656 (>=0), A005838 (>0);
%Y A020655 7-term AP: A020657 (>=0), A020658 (>0);
%Y A020655 8-term AP: A020659 (>=0), A020660 (>0);
%Y A020655 9-term AP: A020661 (>=0), A020662 (>0);
%Y A020655 10-term AP: A020663 (>=0), A020664 (>0).
%K A020655 nonn
%O A020655 1,2
%A A020655 _David W. Wilson_