_Paul D. Hanna (pauldhanna(AT)juno.com), _, Aug 22 2006

Consists entirely of 3's, 5's, and 7's, after an initial partial quotient of 4. These partial quotients are aperiodic.

cofr,nonn,new

Decimal expansion (A122164A078886) consists of large gaps of zeros between strings of digits that form powers of 2; this can be seen by grouping the digits as follows:

Cf. A122164A078886.

cofr,nonn,new

cofr,nonn,new

Paul D . Hanna (pauldhanna(AT)juno.com), Aug 22 2006

Continued fraction expansion of constant x = Sum_{n>=0} 1/5^(2^n).

0, 4, 7, 5, 5, 3, 5, 7, 5, 3, 7, 5, 3, 5, 5, 7, 5, 3, 7, 5, 5, 3, 5, 7, 3, 5, 7, 5, 3, 5, 5, 7, 5, 3, 7, 5, 5, 3, 5, 7, 5, 3, 7, 5, 3, 5, 5, 7, 3, 5, 7, 5, 5, 3, 5, 7, 3, 5, 7, 5, 3, 5, 5, 7, 5, 3, 7, 5, 5, 3, 5, 7, 5, 3, 7, 5, 3, 5, 5, 7, 5, 3, 7, 5, 5, 3, 5, 7, 3, 5, 7, 5, 3, 5, 5, 7, 3, 5, 7, 5, 5, 3, 5, 7, 5

0,2

Consists entirely of 3's, 5's, and 7's, after an initial partial quotient of 4. These partial quotients are aperiodic.

x=[0;4,7,5,5,3,5,7,5,3,7,5,3,5,5,7,5,3,7,5,5,3,5,7,3,5,7,5,3,5,5,7,5,...].

x=0.2416025600065536000000429496729600000000000018446744073709551616000...

Decimal expansion (A122164) consists of large gaps of zeros between strings of digits that form powers of 2; this can be seen by grouping the digits as follows:

x = .2 4 16 0 256 000 65536 000000 4294967296 000000000000 ...

and then recognizing the substrings as powers of 2:

2 = 2^(2^0), 4 = 2^(2^1), 16 = 2^(2^2), 65536 = 2^(2^4),

4294967296 = 2^(2^5), 18446744073709551616 = 2^(2^6), ...

(PARI) {a(n)=local(x=sum(k=0, ceil(3+log(n+1)), 1/5^(2^k))); contfrac(x)[n+1]}

Cf. A122164.

cofr,nonn

Paul D Hanna (pauldhanna(AT)juno.com), Aug 22 2006

approved