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

1,3

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?

Links

Table of n, a(n) for n=1..50.

Example

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.

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

Prog

(PARI) A240556(n, show=0, L=5, o=3, v=[0], D=v->v[2..-1]-v[1..-2])={ my(d, m); while( #v<n, show&&print1(v[#v]", "); v=concat(v, v[#v]); while( v[#v]++, forvec( i=vector(L, j, [if(j<L, j, #v), #v]), d=D(vecextract(v, i)); m=o; while(m--&&#Set(d=D(d))>1, ); #Set(d)>1||next(2), 2); break)); v[#v]} \\ M. F. Hasler, Jan 12 2016

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.

Sequence in context: A209038 A182803 A106532 * A090693 A260794 A277576

Adjacent sequences: A240553 A240554 A240555 * A240557 A240558 A240559

Keyword

nonn

Author

T. D. Noe, Apr 09 2014

Status

approved