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Decimal expansion of the number whose binary expansion differs from its decimal expansion only in the first digit.
1

%I #26 Jul 07 2022 02:01:00

%S 1,0,0,1,1,0,0,1,1,0,1,0,0,0,0,0,1,1,0,0,1,1,1,1,0,1,0,0,0,1,1,1,0,1,

%T 0,0,1,0,1,0,0,1,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0,

%U 0,0,0,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0,0,1,1,0,1,1,1,0,0,0,1,0,1

%N Decimal expansion of the number whose binary expansion differs from its decimal expansion only in the first digit.

%C The decimal fraction 0.1 has binary expansion starting with 0.0001...; copying the suffix 001 (3 digits, as 3 < log_2(10) < 4) we obtain 0.1001, which expands to 0.00011001101, etc.

%C Alternatively the process can be described as greedily expressing 1/2 with digits of weights 1/2^n-1/10^n. With f(n)=1/2^n-1/10^n, 0.5 = f(1)+f(4)+f(5)+f(8)+f(9)+f(11)...

%e 0.100110011010000011001111010001110100101001000111010001001101001011...

%t seq[len_] := Module[{s = Table[0, {len}], x = 1/10, n = 1, c = 0}, s[[1]] = 1; While[n < len, While[1/2^n - 1/10^n > x, n++]; c++; s[[n]] = 1; x -= (1/2^n - 1/10^n)]; s]; seq[100] (* _Amiram Eldar_, Jun 29 2022 *)

%Y Cf. A352677 (golden base = binary).

%K nonn,cons

%O 0,1

%A _Leonid Broukhis_, Jun 29 2022