%I #13 Jul 29 2015 13:50:41

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

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

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

%N Decimal expansion 2+(1/log(Phi))*sum(k>=3,log(((2^k-1)*Phi^2+2*Phi)/((2^k-1)*Phi^2-2)))=3.1152... where Phi=(1+sqrt(5))/2.

%H Brigitte Vallée, <a href="http://www.numdam.org/item?id=JTNB_2000__12_2_531_0">Digits and continuants in Euclidean algorithms. Ergodic vs tauberian theorems</a>, Journal de théorie des nombres de Bordeaux 12 (2000), 519-558.

%t digits = 105; $MaxExtraPrecision = digits + 5; NSum[ Log[ (GoldenRatio^2*(2^k - 1) + 2*GoldenRatio)/((2^k - 1)*GoldenRatio^2 - 2)], {k, 3, Infinity}, NSumTerms -> digits, WorkingPrecision -> digits + 5]/Log[GoldenRatio] + 2 // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Mar 04 2013 *)

%K cons,nonn

%O 1,1

%A _Benoit Cloitre_, Mar 23 2010