A248641 Lexicographically earliest positive sequence which does not contain a 4-term equidistant subsequence (a(n+k*d); k=0,1,2,3) in arithmetic progression.

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, 2, 3, 2, 1, 1, 3, 3, 1, 1, 1

Offset: 0

Author

M. F. Hasler, Oct 10 2014

Keywords

• nonn
• easy

Comments

See A248625 for more information, links and examples.

It is a variation of A229037 where 3-term is replaced by 4-term (and with “lead index” 0 instead of 1)

Links

• Sébastien Palcoux, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A248625, A248639, A248640, A248627, A229037, A241752.

Programs

PARI

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

SageMath

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:
               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
# Sébastien Palcoux, Aug 28 2019