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

Author

Jeffrey Shallit, May 26 2014

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Differs from A094870 in that sequence must be increasing.

Cf. A003278, A229037.

Programs

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 *)