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

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

1,2

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.

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 *)

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).

Sequence in context: A190842 A273106 A191179 * A214980 A098161 A026194

Adjacent sequences: A005834 A005835 A005836 * A005838 A005839 A005840

Keyword

nonn

Author

N. J. A. Sloane, Jeffrey Shallit

Extensions

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

Status

approved