A240556 Earliest nonnegative increasing sequence with no 5-term subsequence of constant third differences.

0, 1, 2, 3, 5, 7, 11, 15, 16, 27, 47, 48, 64, 95, 175, 196, 211, 212, 214, 247, 249, 252, 398, 839, 1002, 1014, 1016, 1035, 1036, 1037, 1051, 1054, 1072, 1121, 1143, 1146, 1172, 1258, 4271, 4282, 4284, 4336, 4571, 4578, 4582, 4598, 4613, 4622, 4628, 4646

Offset: 1

Author

T. D. Noe, Apr 09 2014

Keywords

• nonn

Comments

For the positive sequence, see A240557, which is this sequence plus 1. Is there a simple way of determining this sequence, as in the case of the no 3-term arithmetic progression?

Examples

			After (0, 1, 2, 3, 5, 7), the number 10 is excluded since else the subsequence (0, 2, 3, 5, 10) would have successive 1st, 2nd and 3rd differences (2, 1, 2, 5), (-1, 1, 3) and (2, 2), which is constant and thus excluded.
		

Crossrefs

Cf. A240557 (starting with 1).

No 3-term AP: A005836 (>=0), A003278 (>0);

no 4-term AP: A240075 (>=0), A240555 (>0);

no 5-term AP: A020654 (>=0), A020655 (>0);

no 6-term AP: A020656 (>=0), A005838 (>0);

no 7-term AP: A020657 (>=0), A020658 (>0);

no 8-term AP: A020659 (>=0), A020660 (>0);

no 9-term AP: A020661 (>=0), A020662 (>0);

no 10-term AP: A020663 (>=0), A020664 (>0).

Cf. A240075 and A240555 for sequences avoiding 4-term subsequences with constant second differences.

Programs

Mathematica

t = {0, 1, 2, 3}; Do[s = Table[Append[i, n], {i, Subsets[t, {4}]}]; If[! MemberQ[Flatten[Table[Differences[i, 4], {i, s}]], 0], AppendTo[t, n]], {n, 4, 5000}]; t

PARI

A240556(n,show=0,L=5,o=3,v=[0],D=v->v[2..-1]-v[1..-2])={ my(d,m); while( #v1,);#Set(d)>1||next(2),2);break));v[#v]} \\ M. F. Hasler, Jan 12 2016