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

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, 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, 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

Offset: 1

Author

Sébastien Palcoux, Aug 21 2019

Keywords

• nonn
• look
• easy
• hear

Comments

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).

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).

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

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

Links

• Sébastien Palcoux, Table of n, a(n) for n = 1..100000
• Sébastien Palcoux, On the first sequence without triple in arithmetic progression (version: 2019-08-21), second part, MathOverflow
• Sébastien Palcoux, Table of n, a(n) for n = 1..1000000
• Sébastien Palcoux, Density plot of the first 1000000 terms

Crossrefs

Cf. A003278, A005836, A039966, A229037.

Programs

Python

from itertools import count, islice
def A309890_gen(): # generator of terms
    blist = []
    for n in count(0):
        i, j, b = 1, 1, set()
        while n-(i<<1) >= 0:
            x, y = blist[n-2*i], blist[n-i]
            z = (y<<1)-x
            if x<=y<=z:
                b.add(z)
                while j in b:
                    j += 1
            i += 1
        blist.append(j)
        yield j
A309890_list = list(islice(A309890_gen(),30)) # Chai Wah Wu, Sep 12 2023

SageMath

# %attach SAGE/ThreeFree.spyx
from sage.all import *
cpdef ThreeFree(int n):
     cdef int i,j,k,s,Li,Lj
     cdef list L,Lb
     cdef set b
     L=[1,1]
     for k in range(2,n):
          b=set()
          for i in range(k):
               if 2*((i+k)/2)==i+k:
                    j=(i+k)/2
                    Li,Lj=L[i],L[j]
                    s=2*Lj-Li
                    if s>0 and Li<=Lj:
                         b.add(s)
          if 1 not in b:
               L.append(1)
          else:
               Lb=list(b)
               Lb.sort()
               for t in Lb:
                    if t+1 not in b:
                         L.append(t+1)
                         break
     return L