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

%I A240555 #35 Nov 08 2016 19:48:35
%S A240555 1,2,3,5,6,9,16,17,18,21,45,52,53,54,57,59,65,79,167,168,193,196,197,
%T A240555 201,203,204,207,218,227,249,250,277,313,650,658,679,682,683,716,727,
%U A240555 741,744,748,751,772,791,831,834,837,839,843,855,876,909,912,972
%N A240555 Lexicographically earliest positive increasing sequence such that no four terms have constant second differences.
%C A240555 If "positive" is changed to "nonnegative" we get A240075, which is this sequence minus 1.
%C A240555 See A005837 for the earliest sequence containing no 4-term arithmetic progression.
%H A240555 T. D. Noe, <a href="/A240555/b240555.txt">Table of n, a(n) for n = 1..755</a> (terms < 10^6)
%e A240555 After 1,2,3 the number 4 is excluded since (1,2,3,4) has zero second and third differences.
%e A240555 After 1,2,3,5 the number 8 is excluded since (2,3,5,8) has second differences 1,1.
%t A240555 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
%o A240555 (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)>1||next(2), 2); break)); v[#v]} \\ _M. F. Hasler_, Jan 12 2016
%Y A240555 Summary of increasing sequences avoiding arithmetic progressions of specified lengths (the second of each pair is obtained by adding 1 to the first):
%Y A240555 3-term AP: A005836 (>=0), A003278 (>0);
%Y A240555 4-term AP: A005839 (>=0), A005837 (>0);
%Y A240555 5-term AP: A020654 (>=0), A020655 (>0);
%Y A240555 6-term AP: A020656 (>=0), A005838 (>0);
%Y A240555 7-term AP: A020657 (>=0), A020658 (>0);
%Y A240555 8-term AP: A020659 (>=0), A020660 (>0);
%Y A240555 9-term AP: A020661 (>=0), A020662 (>0);
%Y A240555 10-term AP: A020663 (>=0), A020664 (>0).
%Y A240555 Cf. A240075 (nonnegative version, a(n)-1).
%Y A240555 Cf. A240556 and A240557 for sequences avoiding 5-term subsequences with constant third differences.
%K A240555 nonn
%O A240555 1,2
%A A240555 _T. D. Noe_, Apr 09 2014
%E A240555 Definition corrected by _N. J. A. Sloane_, Jan 04 2016 and _M. F. Hasler_ at the suggestion of Lewis Chen