A248641 -id:A248641 - cp's OEIS frontend

A248639 Least nonnegative sequence which does not contain a 4-term equidistant subsequence (a(n+k*d); k=0,1,2,3) in arithmetic progression.

0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 4, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 0, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 2, 2, 0, 2, 2, 2, 4, 4, 3, 0, 0, 0, 2, 0, 1, 2, 0, 4, 2, 1, 5, 0, 2, 1, 1, 2, 1, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 1, 1, 1, 4, 1, 2, 3, 0, 1, 2, 1, 0, 3, 3, 4, 1, 1, 3

Offset: 0

M. F. Hasler, Oct 10 2014

nonn

See A248625 for more information, links and examples.

See A248641 for the "positive integers" variant.

Cf. A248625, A248640, A248641, A248627, A229037, A241752.

a=[];for(n=1,190,a=concat(a,0);while(hasAP(a,4),a[#a]++));a \\ See A248625 for hasAP().

A322286 Lexicographically earliest sequence of positive integers without 4 terms in a weakly increasing arithmetic progression.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 3, 1, 1, 3, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 2, 3, 3, 5, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 1, 1, 1, 2, 2, 2, 3, 4, 2, 3, 2, 2, 2, 3, 3, 1, 3, 3, 3, 5, 5, 4, 1, 1, 1, 3, 1, 2, 3, 1, 5, 3, 2, 6, 1, 3, 2, 1, 3, 2, 1, 1, 3, 3, 1, 1, 1

Offset: 1

Sébastien Palcoux, Aug 28 2019

This is a variation of A248641 (where we only exclude weakly increasing arithmetic progressions): they differ from the 101st term.
It is also a variation of A309890 where 3-term is replaced by 4-term.
The numbers n for which the n-th term is 1 are given by A005837.
There is no upper bound, because if there were an upper bound r then there must be s <= r such that the set of numbers n for which the n-th term is s has positive density and this contradicts Szemerédi's theorem.
Assuming Erdős's conjecture on arithmetic progressions, for a fixed positive integer r, the sum of the reciprocals of the numbers n for which the n-th term is r converges.

Sébastien Palcoux, Table of n, a(n) for n = 1..10000
Wikipedia, Szemerédi's theorem
Wikipedia, Erdős conjecture on arithmetic progressions

Cf. A005837, A248641, A309890.

cpdef FourFree(int n):
   cdef int i,r,k,s,L1,L2,L3
   cdef list L,Lb
   cdef set b
   L=[1,1,1]
   for k in range(3,n):
      b=set()
      for i in range(k):
         if 3*((k-i)/3)==k-i:
            r=(k-i)/3
            L1,L2,L3=L[i],L[i+r],L[i+2*r]
            s=3*(L2-L1)+L1
            if s>0 and L3==2*(L2-L1)+L1:
               if L1<=L2:
                  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

cp's OEIS Frontend

A248639 Least nonnegative sequence which does not contain a 4-term equidistant subsequence (a(n+k*d); k=0,1,2,3) in arithmetic progression.

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