On Thu, 25 Dec 2008, wouter meeussen wrote:
Googling for 1.5353705088 produces: http://linkinghub.elsevier.com/retrieve/pii/S0898122196002295
and http://books.google.be/books?id=DL5iVYNoEa0C&pg=PA559&lpg=PA559&dq=1.5353705... where we can read Steven Finch's entry: "1.5353705088... With digital search tree constants [5.14]"
Since the number in question is also known to be Sum[1/(Fibonacci[2n]),{n,1,\[Infinity]}]//N Out[]=1.53537050883625298502985289665159900636701159107 as discussed in http://mathworld.wolfram.com/ReciprocalFibonacciConstant.html
it would be worthwile to check if both Finch's and S. Paszkowski's references contain an as yet unknown closed form for the infinite sum.
If you count q-polygamma as "closed" :) In[128]:= Sqrt[5]/(4 Log[GoldenRatio]) (I Pi - QPolyGamma[0, 1, GoldenRatio^2] + QPolyGamma[0, 1 - (I Pi)/(2 Log[GoldenRatio]), GoldenRatio^2]) // N // Chop Out[128]= 1.53537 In[126]:= NSum[1/Fibonacci[2 n], {n, \[Infinity]}] Out[126]= 1.53537 Merry Christmas, -Eric