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Decimal expansion of Sum_{n>0} n/(2^(n^2)).
1

%I #18 Jan 27 2023 08:11:09

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

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

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

%N Decimal expansion of Sum_{n>0} n/(2^(n^2)).

%C This number is obviously 2-dense, but not 2-normal: any finite binary string s representing the value N will appear in its digits, not later than those added by the term N/2^(N^2), but nonzero digits have density zero since the gap between those added by subsequent terms is increasing much faster (~ n) than the maximal possible number of new nonzero digits (~ log_2(n)). - _M. F. Hasler_, Mar 22 2017

%H David H. Bailey and Richard E. Crandall, <a href="http://crd.lbl.gov/~dhbailey/dhbpapers/bcnormal.pdf">Random Generators and Normal Numbers</a>, page 27.

%e 0.6309205592551858647783240039079433700921514299217879868...

%t RealDigits[N[Sum[n/(2^(n^2)), {n, 4!}], 100]][[1]] (* _Arkadiusz Wesolowski_, Sep 29 2011 *)

%o (PARI) default(realprecision,100);sum(n=1,100,n/(2^(n^2)),0.) \\ Typo corrected. sum(n=1,100,n*1.>>(n^2)) is 25 x faster for 1000 digits. - _M. F. Hasler_, Mar 22 2017

%Y Cf. A066716: binary Champernowne constant.

%K cons,nonn

%O 0,1

%A Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004

%E Offset corrected by _Arkadiusz Wesolowski_, Sep 29 2011