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A005837 Lexicographically earliest increasing sequence of positive numbers that contains no 4-term arithmetic progression.
(Formerly M0621)

%I M0621

%S 1,2,3,5,6,8,9,10,15,16,17,19,26,27,29,30,31,34,37,49,50,51,53,54,56,

%T 57,58,63,65,66,67,80,87,88,89,91,94,99,102,105,106,109,110,111,122,

%U 126,136,145,149,151,152,160,163,167,169,170,171,174,176,177,183,187,188,194,196

%N Lexicographically earliest increasing sequence of positive numbers that contains no 4-term arithmetic progression.

%C a(n) = A005839(n) + 1. - _Alois P. Heinz_, Jan 31 2014

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz and Robert Israel, <a href="/A005837/b005837.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..1001 from Alois P. Heinz)

%H J. L. Gerver and L. T. Ramsey, <a href="http://dx.doi.org/10.1090/S0025-5718-1979-0537982-0">Sets of integers with no long arithmetic progressions generated by the greedy algorithm</a>, Math. Comp., 33 (1979), 1353-1359.

%p Noap:= proc(N,m)

%p # N terms of earliest increasing seq with no m-term arithmetic progression

%p local A,forbid,n,c,ds,j;

%p A:= Vector(N):

%p A[1..m-1]:= <($1..m-1)>:

%p forbid:= {m}:

%p for n from m to N do

%p c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);

%p A[n]:= c;

%p ds:= convert(map(t -> c-t, A[m-2..n-1]),set);

%p for j from m-2 to 2 by -1 do

%p ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]),set);

%p if ds = {} then break fi;

%p od;

%p forbid:= select(`>`,forbid,c) union map(`+`,ds,c);

%p od:

%p convert(A,list)

%p end proc:

%p Noap(100,4); # _Robert Israel_, Jan 04 2016

%t t = {1, 2, 3}; Do[s = Table[Append[i, n], {i, Subsets[t, {3}]}]; If[! MemberQ[Table[Differences[i, 2], {i, s}], {0, 0}], AppendTo[t, n]], {n, 4, 200}]; t (* _T. D. Noe_, Apr 17 2014 *)

%Y Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):

%Y 3-term AP: A005836 (>=0), A003278 (>0);

%Y 4-term AP: A005839 (>=0), A005837 (>0);

%Y 5-term AP: A020654 (>=0), A020655 (>0);

%Y 6-term AP: A020656 (>=0), A005838 (>0);

%Y 7-term AP: A020657 (>=0), A020658 (>0);

%Y 8-term AP: A020659 (>=0), A020660 (>0);

%Y 9-term AP: A020661 (>=0), A020662 (>0);

%Y 10-term AP: A020663 (>=0), A020664 (>0).

%K nonn

%O 1,2

%A _N. J. A. Sloane_, _Jeffrey Shallit_

%E Edited by _M. F. Hasler_, Jan 03 2016. Further edited (with new offset) by _N. J. A. Sloane_, Jan 04 2016

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Last modified February 22 15:52 EST 2019. Contains 320399 sequences. (Running on oeis4.)