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A380373 Decimal expansion of Sum_{i>=1} 1/2^A082851(i).

%I A380373 #37 Feb 09 2025 14:32:41
%S A380373 8,6,4,1,9,1,3,2,1,4,9,5,0,4,5,8,6,2,8,7,8,4,6,5,4,8,0,5,8,7,7,0,4,8,
%T A380373 0,2,0,2,3,8,5,1,8,9,1,9,2,8,6,1,4,3,2,0,5,0,6,7,0,2,4,2,4,3,6,4,3,9,
%U A380373 1,7,8,8,7,0,8,5,9,3,2,7,2,0,2,5,8,0,9,0,9,6,3,9,2,7,6,2,1,0,2,3,2,0,9,0,8,3,1,5
%N A380373 Decimal expansion of Sum_{i>=1} 1/2^A082851(i).
%C A380373 This number has the property that the geometric mean of the differences (A082850) in the positions of 1s (A082851) of its binary representation (A380372) approaches the Somos constant (A112302).
%H A380373 Wikipedia, <a href="https://en.wikipedia.org/wiki/Somos%27_quadratic_recurrence_constant#Universality">Somos Constant</a>
%F A380373 Equals Sum_{i>=1} 1/2^A082851(i).
%F A380373 Equals Sum_{i>=1} A380372(i)/2^i.
%e A380373 0.8641913214950458_10 -> 0.1101110100111011101_2 (A380372)
%e A380373 Positions of 1s -> 1,2,4,5,6,8,11,12,13,15,... (A082851)
%e A380373 Difference in positions of 1s -> 1,1,2,1,1,2,3,1,1,2,1,1,2,3,4,... (A082850)
%e A380373 Geomtric Mean -> 1.66168794963359... (A112302)
%o A380373 (Python)
%o A380373 from itertools import count, islice
%o A380373 from fractions import Fraction
%o A380373 import os
%o A380373 def A380372_gen():
%o A380373     S = []
%o A380373     for n in count(1):
%o A380373         yield from (m:=S+[0]*(n-1)+[1])
%o A380373         S += m
%o A380373 def bin_to_frac_interval(binary_repr):
%o A380373     lower_bound, last_bit_id = 0, 0
%o A380373     for i, bit in enumerate(binary_repr, start=1):
%o A380373         if bit:
%o A380373             lower_bound += Fraction(1, 2**i)
%o A380373             last_bit_id = i
%o A380373     upper_bound = lower_bound + Fraction(1, 2**last_bit_id)
%o A380373     return lower_bound, upper_bound
%o A380373 n_binary_terms = 400
%o A380373 diff_bin = islice(A380372_gen(), n_binary_terms)
%o A380373 lower, upper = bin_to_frac_interval(diff_bin)
%o A380373 lower, upper = str(int(lower*10**(n_binary_terms//3))), str(int(upper*10**(n_binary_terms//3)))
%o A380373 A380373 = os.path.commonprefix([lower, upper])
%Y A380373 Cf. A082850, A082851, A380372 (binary expansion).
%K A380373 nonn,cons
%O A380373 0,1
%A A380373 _Jwalin Bhatt_, Jan 23 2025