A240555 Lexicographically earliest positive increasing sequence such that no four terms have constant second differences.

1, 2, 3, 5, 6, 9, 16, 17, 18, 21, 45, 52, 53, 54, 57, 59, 65, 79, 167, 168, 193, 196, 197, 201, 203, 204, 207, 218, 227, 249, 250, 277, 313, 650, 658, 679, 682, 683, 716, 727, 741, 744, 748, 751, 772, 791, 831, 834, 837, 839, 843, 855, 876, 909, 912, 972

Offset: 1

Author

T. D. Noe, Apr 09 2014

Keywords

• nonn

Comments

If "positive" is changed to "nonnegative" we get A240075, which is this sequence minus 1.

See A005837 for the earliest sequence containing no 4-term arithmetic progression.

Examples

			After 1,2,3 the number 4 is excluded since (1,2,3,4) has zero second and third differences.
After 1,2,3,5 the number 8 is excluded since (2,3,5,8) has second differences 1,1.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..755 (terms < 10^6)

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).

Cf. A240075 (nonnegative version, a(n)-1).

Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.

Programs

Mathematica

t = {1, 2, 3}; Do[s = Table[Append[i, n], {i, Subsets[t, {3}]}]; If[! MemberQ[Flatten[Table[Differences[i, 3], {i, s}]], 0], AppendTo[t, n]], {n, 4, 1000}]; t

PARI

A240555(n, show=0, L=4, o=2, v=[1], 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

Extensions

Definition corrected by N. J. A. Sloane, Jan 04 2016 and M. F. Hasler at the suggestion of Lewis Chen