OFFSET
1,1
COMMENTS
LINKS
Levi van de Pol, The first occurrence of a number in Gijswijt's sequence, arXiv:2209.04657 [math.CO], 2022.
FORMULA
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".
EXAMPLE
3.48669886438365597023...
PROG
(Python)
import math
from mpmath import *
# warning: 0.1 and mpf(1/10) are incorrect. Use mpf(1)/mpf(10)
mp.dps=60
def Cn(X):
l=len(X)
cn=1
for i in range(1, int(l/2)+1):
j=i
while(X[l-j-1]==X[l-j-1+i]):
j=j+1
if j>=l:
break
candidate=int(j/i)
if candidate>cn:
cn=candidate
return cn
def epsilon():
A=[2] # level-2 Gijswijt sequence
number=1 # number of S strings encountered
position=0 # position of end of last S
value=mpf(1) # approximation for epsilon1
for i in range(1, 6000):
k=Cn(A)
A.append(max(2, k))
if k<2:
value=value+mpf(i-position)/mpf(2**number)
position=mpf(i)
number+=1
return value
print("epsilon_1: ", epsilon())
CROSSREFS
KEYWORD
AUTHOR
Levi van de Pol, Oct 22 2022
STATUS
approved