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

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, 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, 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

Offset

0,1

Comments

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).

Links

Table of n, a(n) for n=0..107.

Wikipedia, Somos Constant

Formula

Equals Sum_{i>=1} 1/2^A082851(i).

Equals Sum_{i>=1} A380372(i)/2^i.

Example

0.8641913214950458_10 -> 0.1101110100111011101_2 (A380372)
Positions of 1s -> 1,2,4,5,6,8,11,12,13,15,... (A082851)
Difference in positions of 1s -> 1,1,2,1,1,2,3,1,1,2,1,1,2,3,4,... (A082850)
Geomtric Mean -> 1.66168794963359... (A112302)

Prog

(Python)
from itertools import count, islice
from fractions import Fraction
import os
def A380372_gen():
    S = []
    for n in count(1):
        yield from (m:=S+[0]*(n-1)+[1])
        S += m
def bin_to_frac_interval(binary_repr):
    lower_bound, last_bit_id = 0, 0
    for i, bit in enumerate(binary_repr, start=1):
        if bit:
            lower_bound += Fraction(1, 2**i)
            last_bit_id = i
    upper_bound = lower_bound + Fraction(1, 2**last_bit_id)
    return lower_bound, upper_bound
n_binary_terms = 400
diff_bin = islice(A380372_gen(), n_binary_terms)
lower, upper = bin_to_frac_interval(diff_bin)
lower, upper = str(int(lower*10**(n_binary_terms//3))), str(int(upper*10**(n_binary_terms//3)))
A380373 = os.path.commonprefix([lower, upper])

Crossrefs

Cf. A082850, A082851, A380372 (binary expansion).

Sequence in context: A263181 A298978 A199151 * A343275 A177154 A059631

Adjacent sequences: A380370 A380371 A380372 * A380374 A380375 A380376

Keyword

nonn,cons

Author

Jwalin Bhatt, Jan 23 2025

Status

approved