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

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, 37, 38, 39, 41, 42, 43, 44, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94, 126, 127, 128, 129, 131, 132, 133

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..10000

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, 5); # Robert Israel, Jan 04 2016

Mathematica

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 *)

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: A101888 A190241 A225859 * A188042 A236842 A236860

Adjacent sequences: A020652 A020653 A020654 * A020656 A020657 A020658

Keyword

nonn

Author

David W. Wilson

Status

approved