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A383671 The limiting word that starts with 0, as a sequence, generated by s(0) = 0, s(1) = 12, s(n) = concatenation of s(n - 2) and s(n - 1).

%I A383671 #18 May 21 2025 23:36:34
%S A383671 0,1,2,1,2,0,1,2,1,2,0,1,2,0,1,2,1,2,0,1,2,1,2,0,1,2,0,1,2,1,2,0,1,2,
%T A383671 0,1,2,1,2,0,1,2,1,2,0,1,2,0,1,2,1,2,0,1,2,1,2,0,1,2,0,1,2,1,2,0,1,2,
%U A383671 0,1,2,1,2,0,1,2,1,2,0,1,2,0,1,2,1,2
%N A383671 The limiting word that starts with 0, as a sequence, generated by s(0) = 0, s(1) = 12, s(n) = concatenation of s(n - 2) and s(n - 1).
%C A383671 There are two distinct limiting words generated by s(0) = 0, s(1) = 12, s(n) = s(n - 2)s(n - 1). This one is given by s(2n) for n>=0; the other, given by s(2n-1) for n>=0, is 1201201212012... In both limiting words, the length of the n-th initial subword is A000045(n+1), for n>=1.
%e A383671 Initial subwords: s(0)=0, s(1)=12, s(2)=012, s(3)=12012, s(4)= 01212012, of lengths 1, 2, 3, 5, 8 (Fibonacci numbers).
%t A383671 s[0] = "0"; s[1] = "12"; s[n_] := StringJoin[s[n - 2], s[n - 1]];
%t A383671 Join[{0}, IntegerDigits[FromDigits[s[10]]]]
%o A383671 (Python)
%o A383671 from math import isqrt
%o A383671 def A047924(n): return ((m:=(n+isqrt(5*n**2)>>1)+1)+isqrt(5*m**2)>>1)+m+1
%o A383671 def A026356(n): return (n+1+isqrt(5*(n-1)**2)>>1)+n
%o A383671 def A383671(n):
%o A383671     def bsearch(f, n):
%o A383671         kmin, kmax = 0, 1
%o A383671         while f(kmax) <= n:
%o A383671             kmax <<= 1
%o A383671         kmin = kmax>>1
%o A383671         while True:
%o A383671             kmid = kmax+kmin>>1
%o A383671             if f(kmid) > n:
%o A383671                 kmax = kmid
%o A383671             else:
%o A383671                 kmin = kmid
%o A383671             if kmax-kmin <= 1:
%o A383671                 break
%o A383671         return kmin
%o A383671     if n<3: return n
%o A383671     for i, f in enumerate((A047924, A026356)):
%o A383671         if f(bsearch(f,n+1))==n+1: return i
%o A383671     return 2 # _Chai Wah Wu_, May 21 2025
%Y A383671 Cf. A000045, A003849, A047924 (positions of 0), A026356 (positions of 1), A022413 (positions of 2), A383670.
%K A383671 nonn
%O A383671 0,3
%A A383671 _Clark Kimberling_, May 15 2025