|
%I #23 Feb 27 2018 16:29:07
%S 0,1,2,3,5,7,11,15,16,27,47,48,64,95,175,196,211,212,214,247,249,252,
%T 398,839,1002,1014,1016,1035,1036,1037,1051,1054,1072,1121,1143,1146,
%U 1172,1258,4271,4282,4284,4336,4571,4578,4582,4598,4613,4622,4628,4646
%N Earliest nonnegative increasing sequence with no 5-term subsequence of constant third differences.
%C 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?
%e 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.
%t 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
%o (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
%Y Cf. A240557 (starting with 1).
%Y No 3-term AP: A005836 (>=0), A003278 (>0);
%Y no 4-term AP: A240075 (>=0), A240555 (>0);
%Y no 5-term AP: A020654 (>=0), A020655 (>0);
%Y no 6-term AP: A020656 (>=0), A005838 (>0);
%Y no 7-term AP: A020657 (>=0), A020658 (>0);
%Y no 8-term AP: A020659 (>=0), A020660 (>0);
%Y no 9-term AP: A020661 (>=0), A020662 (>0);
%Y no 10-term AP: A020663 (>=0), A020664 (>0).
%Y Cf. A240075 and A240555 for sequences avoiding 4-term subsequences with constant second differences.
%K nonn
%O 1,3
%A _T. D. Noe_, Apr 09 2014
|
