A205325 Decimal expansion of the limit of [0;1,1,...] + [0;2,2,...] + ... + [0;n,n,...] - log(n) as n approaches infinity.

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

Offset

0,2

Links

Table of n, a(n) for n=0..87.

Martin Janecke, Edle Reihe

Formula

lim_{n->infinity} (1/[1;1,...] + 1/[2;2,...] + 1/[3;3,...] + ... + 1/[n;n,...] - log(n)).

lim_{n->infinity} (sum_{k=1...n} (2/(k + sqrt(k^2 + 4))) - log(n)).

Example

0.0416662....

Mathematica

digits = 10; dn = 1000000; Clear[f]; f[n_] := NSum[2/(k + Sqrt[k^2+4]) - 1/k, {k, 1, Infinity}, NSumTerms -> 200000, WorkingPrecision -> digits+10, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 20}}] + EulerGamma // RealDigits[#, 10, digits+2]& // First; f[dn]; f[n = 2*dn]; While[f[n] != f[n-dn], n = n+dn]; Prepend[ f[n][[1 ;; digits]], 0] (* Jean-François Alcover, Feb 25 2013 *)

Crossrefs

Cf. A001620, A205326, continued fractions A001622, A014176, A098316, A098317, A098318.

Sequence in context: A011242 A371498 A008565 * A021100 A021028 A193529

Adjacent sequences: A205322 A205323 A205324 * A205326 A205327 A205328

Keyword

cons,nonn

Author

Martin Janecke, Jan 26 2012

Extensions

More terms from Jean-François Alcover, Feb 25 2013

More terms from Jon E. Schoenfield, Jan 05 2014

Status

approved