A242921 Lexicographically least increasing sequence avoiding double 3-term arithmetic progressions.

0, 1, 3, 4, 7, 8, 10, 11, 15, 17, 18, 20, 25, 27, 28, 31, 32, 34, 35, 38, 42, 43, 45, 46, 53, 55, 58, 59, 61, 62, 67, 68, 70, 71, 79, 81, 85, 87, 90, 92, 93, 98, 102, 105, 112, 114, 115, 119, 121, 126, 129, 130, 132, 133, 136, 140, 141, 143, 144, 148

Offset

0,3

Comments

a(0) = 0, a(1) = 1, and for n >= 2, a(n) is the least integer t > a(n-1) such that for all 0 < i <= n/2 we have a(n-2i)+t <> 2a(n-i).

By double arithmetic sequence it is meant that both the indices and the values are in arithmetic progression.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

T. Brown, V. Jungic, and A. Poelstra, On double 3-term arithmetic progressions, arxiv preprint, November 2013.

Example

a(8) = 15: 12 is not in the sequence because a(6) = 10, a(7) = 11; 13 is not in the sequence because a(4) = 7, a(6) = 10; 14 is not in the sequence because a(0) = 0, a(4) = 7, so a(8) = 15.

Maple

a:= proc(n) option remember; local i, t, ok;
      if n<2 then n
    else for t from 1+a(n-1) do ok:=true;
           for i to n/2 while ok
             do ok:=a(n-2*i)+t <> 2*a(n-i) od;
           if ok then return t fi
         od
      fi
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 26 2014

Mathematica

a[n_] := a[n] = Module[{i, t, ok}, If[n<2, n, For[t = 1+a[n-1], True, t++, ok = True; i = 1; While[ok && i <= n/2, ok = a[n-2*i]+t != 2*a[n-i]; i++]; If[ok, Return[t]]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 09 2017, after Alois P. Heinz *)

Crossrefs

Differs from A094870 in that sequence must be increasing.

Cf. A003278, A229037.

Sequence in context: A005187 A184835 A184823 * A091934 A178482 A284658

Adjacent sequences: A242918 A242919 A242920 * A242922 A242923 A242924

Keyword

nonn

Author

Jeffrey Shallit, May 26 2014

Status

approved