cp's OEIS Frontend

A357067 Decimal expansion of the limit of A091411(k)/2^(k-1) as k goes to infinity.

%I A357067 #23 Feb 28 2023 09:23:57
%S A357067 3,4,8,6,6,9,8,8,6,4,3,8,3,6,5,5,9,7,0,2,3,5,8,7,2,7,0,0,7,0,2,2,2,0,
%T A357067 6,6,7,3,3,5,4,1,3,6,6,2
%N A357067 Decimal expansion of the limit of A091411(k)/2^(k-1) as k goes to infinity.
%C A357067 In the article "The first occurrence of a number in Gijswijt's sequence", this constant is called epsilon_1. Its existence is proved in Theorem 7.2. The constant occurs in a direct formula (Theorem 7.11) for A091409(n), the first occurrence of the integer n in Gijswijt's sequence A090822.
%H A357067 Levi van de Pol, <a href="https://arxiv.org/abs/2209.04657">The first occurrence of a number in Gijswijt's sequence</a>, arXiv:2209.04657 [math.CO], 2022.
%F A357067 Equal to 1 + Sum_{k>=1} A091579(k)/2^k. Proved in Corollary 7.3 of the article "The first occurrence of a number in Gijswijt's sequence".
%e A357067 3.48669886438365597023...
%o A357067 (Python)
%o A357067 import math
%o A357067 from mpmath import *
%o A357067 # warning: 0.1 and mpf(1/10) are incorrect. Use mpf(1)/mpf(10)
%o A357067 mp.dps=60
%o A357067 def Cn(X):
%o A357067     l=len(X)
%o A357067     cn=1
%o A357067     for i in range(1, int(l/2)+1):
%o A357067         j=i
%o A357067         while(X[l-j-1]==X[l-j-1+i]):
%o A357067             j=j+1
%o A357067             if j>=l:
%o A357067                 break
%o A357067         candidate=int(j/i)
%o A357067         if candidate>cn:
%o A357067             cn=candidate
%o A357067     return cn
%o A357067 def epsilon():
%o A357067     A=[2] # level-2 Gijswijt sequence
%o A357067     number=1 # number of S strings encountered
%o A357067     position=0 # position of end of last S
%o A357067     value=mpf(1) # approximation for epsilon1
%o A357067     for i in range(1,6000):
%o A357067         k=Cn(A)
%o A357067         A.append(max(2,k))
%o A357067         if k<2:
%o A357067             value=value+mpf(i-position)/mpf(2**number)
%o A357067             position=mpf(i)
%o A357067             number+=1
%o A357067     return value
%o A357067 print("epsilon_1: ",epsilon())
%Y A357067 Cf. A090822, A091409, A091411, A091579, A357066, A357068.
%K A357067 nonn,cons,more
%O A357067 1,1
%A A357067 _Levi van de Pol_, Oct 22 2022