A329810 - OEIS

%I #18 Feb 04 2020 21:31:27

%S 7,5,8,5,4,2,3,0,8,1,7,1,0,5,5,7,3,9,2,6,8,1,2,6,0,4,8,8,4,2,2,4,8,8,

%T 9,3,4,2,1,2,4,7,7,7,9,7,9,6,9,5,2,8,6,0,2,9,9,5,5,2,3,9,4,0,3,1,9,0,

%U 9,5,3,5,0,9,0,9,4,0,6,7,2,3,0,8,5,9,8

%N Decimal expansion of the constant whose continued fraction representation is [0; 1, 3, 7, 15, 31, ...] = A000225 (the Mersenne numbers).

%C Since Mersenne numbers of the form 2^x - 1 consist entirely of 1's when written in binary, this continued fraction is nothing but 1's if written in binary.

%C Binary continued fraction: 1/(1+1/(11+1/(111+1/(1111+1/(11111+1/(111111+1/...

%e 0.758542308171055739268126048842248893421247779...

%t N[FromContinuedFraction[Table[2^k - 1, {k, 0, 100}]], 120] (* _Vaclav Kotesovec_, Nov 21 2019 *)

%o (PARI) dec_exp(v)= w=contfracpnqn(v); w[1, 1]/w[2, 1]+0.

%o dec_exp(vector(200, i, 2^(i-1)-1)) \\ _Michel Marcus_, Nov 21 2019

%Y Cf. A000225, A052119, A073824.

%K nonn,cons

%O 0,1

%A _Daniel Hoyt_, Nov 21 2019