A205326 - OEIS

%I #14 Mar 31 2018 14:47:47

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

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

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

%N Decimal expansion of the sum of [0;n,n,n,...]^2 for n=1..infinity.

%C This is the total area of all squares with sides parallel to the axes of the Cartesian coordinate system, the lower left vertex at (n,0) and the upper right vertex on f(x)=1/x for n=1..infinity.

%H Martin Janecke, <a href="http://prlbr.de/2010/01/edle-reihe/">Edle Reihe</a>

%F Sum_{n>=1} 1/[n;n,n,...]^2.

%F Sum_{n>=1} 4/(n + sqrt(n^2 + 4))^2.

%e 0.9155879199018197251998168538031900897353...

%o (PARI) zeta(2)+sumpos(n=1,4/(n+sqrt(n^2+4))^2-1/n^2) \\ _Charles R Greathouse IV_, Jan 26 2012

%Y Cf. A013661, A205325, continued fractions A001622, A014176, A098316, A098317, A098318.

%K cons,nonn

%O 0,1

%A _Martin Janecke_, Jan 26 2012

%E a(-5)-a(-86) from _Charles R Greathouse IV_, Jan 26 2012