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

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, 57, 58, 63, 65, 66, 67, 80, 87, 88, 89, 91, 94, 99, 102, 105, 106, 109, 110, 111, 122, 126, 136, 145, 149, 151, 152, 160, 163, 167, 169, 170, 171, 174, 176, 177, 183, 187, 188, 194, 196

Offset: 1

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• nonn

Comments

a(n) = A005839(n) + 1. - Alois P. Heinz, Jan 31 2014

References

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

Links

• Alois P. Heinz and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..1001 from Alois P. Heinz)
• J. L. Gerver and L. T. Ramsey, Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comp., 33 (1979), 1353-1359.

Crossrefs

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

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

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

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

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

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

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

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

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

Programs

Maple

Noap:= proc(N,m)
# N terms of earliest increasing seq with no m-term arithmetic progression
local A,forbid,n,c,ds,j;
A:= Vector(N):
A[1..m-1]:= <($1..m-1)>:
forbid:= {m}:
for n from m to N do
  c:= min({$A[n-1]+1..max(max(forbid)+1, A[n-1]+1)} minus forbid);
  A[n]:= c;
  ds:= convert(map(t -> c-t, A[m-2..n-1]),set);
  for j from m-2 to 2 by -1 do
    ds:= ds intersect convert(map(t -> (c-t)/j, A[m-j-1..n-j]),set);
    if ds = {} then break fi;
  od;
  forbid:= select(`>`,forbid,c) union map(`+`,ds,c);
od:
convert(A,list)
end proc:
Noap(100,4); # Robert Israel, Jan 04 2016

Mathematica

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 *)

Extensions

Edited by M. F. Hasler, Jan 03 2016. Further edited (with new offset) by N. J. A. Sloane, Jan 04 2016