

A240555


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


12



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
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OFFSET

1,2


COMMENTS

If "positive" is changed to "nonnegative" we get A240075, which is this sequence minus 1.
See A005837 for the earliest sequence containing no 4term arithmetic progression.


LINKS

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


EXAMPLE

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.


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


PROG

(PARI) A240555(n, show=0, L=4, o=2, v=[1], 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)>1next(2), 2); break)); v[#v]} \\ M. F. Hasler, Jan 12 2016


CROSSREFS

Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
3term AP: A005836 (>=0), A003278 (>0);
4term AP: A005839 (>=0), A005837 (>0);
5term AP: A020654 (>=0), A020655 (>0);
6term AP: A020656 (>=0), A005838 (>0);
7term AP: A020657 (>=0), A020658 (>0);
8term AP: A020659 (>=0), A020660 (>0);
9term AP: A020661 (>=0), A020662 (>0);
10term AP: A020663 (>=0), A020664 (>0).
Cf. A240075 (nonnegative version, a(n)1).
Cf. A240556 and A240557 for sequences avoiding 5term subsequences with constant third differences.
Sequence in context: A294631 A059454 A138538 * A102694 A124253 A256230
Adjacent sequences: A240552 A240553 A240554 * A240556 A240557 A240558


KEYWORD

nonn


AUTHOR

T. D. Noe, Apr 09 2014


EXTENSIONS

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


STATUS

approved



