%I #21 Jan 17 2016 09:28:55

%S 0,1,2,4,5,8,15,16,17,20,44,51,52,53,56,58,64,78,166,167,192,195,196,

%T 200,202,203,206,217,226,248,249,276,312,649,657,678,681,682,715,726,

%U 740,743,747,750,771,790,830,833,836,838,842,854,875,908,911,971

%N Lexicographically earliest nonnegative increasing sequence such that no four terms have constant second differences.

%H Vincenzo Librandi and T. D. Noe, <a href="/A240075/b240075.txt">Table of n, a(n) for n = 1..755</a> (first 123 terms from Vincenzo Librandi)

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

%o (PARI) A240075(n, show=0, L=4, o=2, 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

%Y For the positive sequence, see A240555, which is this sequence plus 1.

%Y Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):

%Y 3-term AP: A005836 (>=0), A003278 (>0);

%Y 4-term AP: A005839 (>=0), A005837 (>0);

%Y 5-term AP: A020654 (>=0), A020655 (>0);

%Y 6-term AP: A020656 (>=0), A005838 (>0);

%Y 7-term AP: A020657 (>=0), A020658 (>0);

%Y 8-term AP: A020659 (>=0), A020660 (>0);

%Y 9-term AP: A020661 (>=0), A020662 (>0);

%Y 10-term AP: A020663 (>=0), A020664 (>0).

%Y For the analog sequence which avoids 5-term subsequences of constant third differences, see A240556 (>=0) and A240557 (>0).

%K nonn

%O 1,3

%A _T. D. Noe_, Apr 09 2014

%E Definition corrected by _N. J. A. Sloane_ and _M. F. Hasler_, Jan 04 2016.