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A122165 Continued fraction expansion of constant x = Sum_{n>=0} 1/5^(2^n).

%I A122165 #5 Mar 30 2012 18:36:59
%S A122165 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,
%T A122165 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,
%U A122165 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
%N A122165 Continued fraction expansion of constant x = Sum_{n>=0} 1/5^(2^n).
%C A122165 Consists entirely of 3's, 5's and 7's, after an initial partial quotient of 4. These partial quotients are aperiodic.
%e A122165 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,...].
%e A122165 x=0.2416025600065536000000429496729600000000000018446744073709551616000...
%e A122165 Decimal expansion (A078886) 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:
%e A122165 x = .2 4 16 0 256 000 65536 000000 4294967296 000000000000 ...
%e A122165 and then recognizing the substrings as powers of 2:
%e A122165 2 = 2^(2^0), 4 = 2^(2^1), 16 = 2^(2^2), 65536 = 2^(2^4),
%e A122165 4294967296 = 2^(2^5), 18446744073709551616 = 2^(2^6), ...
%o A122165 (PARI) {a(n)=local(x=sum(k=0,ceil(3+log(n+1)),1/5^(2^k)));contfrac(x)[n+1]}
%Y A122165 Cf. A078886.
%K A122165 cofr,nonn
%O A122165 0,2
%A A122165 _Paul D. Hanna_, Aug 22 2006