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A309890 Lexicographically earliest sequence of positive integers without triples in weakly increasing arithmetic progression.

%I A309890 #65 Sep 12 2023 12:02:09
%S A309890 1,1,2,1,1,2,2,4,4,1,1,2,1,1,2,2,4,4,2,4,4,5,5,8,5,5,9,1,1,2,1,1,2,2,
%T A309890 4,4,1,1,2,1,1,2,2,4,4,2,4,4,5,5,8,5,5,9,2,4,4,5,5,10,5,5,10,10,11,13,
%U A309890 10,11,10,11,13,10,10,12,13,10,13,11,12,20,11,1,1,2,1,1,2,2,4,4,1,1,2,1,1,2,2,4,4,2
%N A309890 Lexicographically earliest sequence of positive integers without triples in weakly increasing arithmetic progression.
%C A309890 Formal definition: lexicographically earliest sequence of positive integers a(n) such that for any i > 0, there is no n > 0 such that 2a(n+i) = a(n) + a(n+2i) AND a(n) <= a(n+i) <= a(n+2i).
%C A309890 Sequence suggested by _Richard Stanley_ as a variant of A229037. They differ from the 55th term. The numbers n for which a(n) = 1 are given by A003278, or equally by A005836 (_Richard Stanley_).
%C A309890 The sequence defined by c(n) = 1 if a(n) = 1 and otherwise c(n) = 0 is A039966 (although with a different offset). - _N. J. A. Sloane_, Dec 01 2019
%C A309890 Pleasant to listen to (button above) with Instrument no. 13: Marimba (and for better listening, save and convert to MP3).
%H A309890 Sébastien Palcoux, <a href="/A309890/b309890.txt">Table of n, a(n) for n = 1..100000</a>
%H A309890 Sébastien Palcoux, <a href="https://mathoverflow.net/q/338415">On the first sequence without triple in arithmetic progression</a> (version: 2019-08-21), second part, MathOverflow
%H A309890 Sébastien Palcoux, <a href="/A309890/a309890.txt">Table of n, a(n) for n = 1..1000000</a>
%H A309890 Sébastien Palcoux, <a href="/A309890/a309890.png">Density plot of the first 1000000 terms</a>
%o A309890 (SageMath)
%o A309890 # %attach SAGE/ThreeFree.spyx
%o A309890 from sage.all import *
%o A309890 cpdef ThreeFree(int n):
%o A309890      cdef int i,j,k,s,Li,Lj
%o A309890      cdef list L,Lb
%o A309890      cdef set b
%o A309890      L=[1,1]
%o A309890      for k in range(2,n):
%o A309890           b=set()
%o A309890           for i in range(k):
%o A309890                if 2*((i+k)/2)==i+k:
%o A309890                     j=(i+k)/2
%o A309890                     Li,Lj=L[i],L[j]
%o A309890                     s=2*Lj-Li
%o A309890                     if s>0 and Li<=Lj:
%o A309890                          b.add(s)
%o A309890           if 1 not in b:
%o A309890                L.append(1)
%o A309890           else:
%o A309890                Lb=list(b)
%o A309890                Lb.sort()
%o A309890                for t in Lb:
%o A309890                     if t+1 not in b:
%o A309890                          L.append(t+1)
%o A309890                          break
%o A309890      return L
%o A309890 (Python)
%o A309890 from itertools import count, islice
%o A309890 def A309890_gen(): # generator of terms
%o A309890     blist = []
%o A309890     for n in count(0):
%o A309890         i, j, b = 1, 1, set()
%o A309890         while n-(i<<1) >= 0:
%o A309890             x, y = blist[n-2*i], blist[n-i]
%o A309890             z = (y<<1)-x
%o A309890             if x<=y<=z:
%o A309890                 b.add(z)
%o A309890                 while j in b:
%o A309890                     j += 1
%o A309890             i += 1
%o A309890         blist.append(j)
%o A309890         yield j
%o A309890 A309890_list = list(islice(A309890_gen(),30)) # _Chai Wah Wu_, Sep 12 2023
%Y A309890 Cf. A003278, A005836, A039966, A229037.
%K A309890 nonn,look,easy,hear
%O A309890 1,3
%A A309890 _Sébastien Palcoux_, Aug 21 2019