A309890 - OEIS

%I #65 Sep 12 2023 12:02:09

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

%C 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 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 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 Pleasant to listen to (button above) with Instrument no. 13: Marimba (and for better listening, save and convert to MP3).

%H Sébastien Palcoux, <a href="/A309890/b309890.txt">Table of n, a(n) for n = 1..100000</a>

%H 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 Sébastien Palcoux, <a href="/A309890/a309890.txt">Table of n, a(n) for n = 1..1000000</a>

%H Sébastien Palcoux, <a href="/A309890/a309890.png">Density plot of the first 1000000 terms</a>

%o (SageMath)

%o # %attach SAGE/ThreeFree.spyx

%o from sage.all import *

%o cpdef ThreeFree(int n):

%o      cdef int i,j,k,s,Li,Lj

%o      cdef list L,Lb

%o      cdef set b

%o      L=[1,1]

%o      for k in range(2,n):

%o           b=set()

%o           for i in range(k):

%o                if 2*((i+k)/2)==i+k:

%o                     j=(i+k)/2

%o                     Li,Lj=L[i],L[j]

%o                     s=2*Lj-Li

%o                     if s>0 and Li<=Lj:

%o                          b.add(s)

%o           if 1 not in b:

%o                L.append(1)

%o           else:

%o                Lb=list(b)

%o                Lb.sort()

%o                for t in Lb:

%o                     if t+1 not in b:

%o                          L.append(t+1)

%o                          break

%o      return L

%o (Python)

%o from itertools import count, islice

%o def A309890_gen(): # generator of terms

%o     blist = []

%o     for n in count(0):

%o         i, j, b = 1, 1, set()

%o         while n-(i<<1) >= 0:

%o             x, y = blist[n-2*i], blist[n-i]

%o             z = (y<<1)-x

%o             if x<=y<=z:

%o                 b.add(z)

%o                 while j in b:

%o                     j += 1

%o             i += 1

%o         blist.append(j)

%o         yield j

%o A309890_list = list(islice(A309890_gen(),30)) # _Chai Wah Wu_, Sep 12 2023

%Y Cf. A003278, A005836, A039966, A229037.

%K nonn,look,easy,hear

%O 1,3

%A _Sébastien Palcoux_, Aug 21 2019