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Decimal expansion of the continued fraction 1/(Pi+1/(Pi+1/(Pi+1/(Pi+...)))).
1

%I #32 Jul 04 2017 17:01:11

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

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

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

%N Decimal expansion of the continued fraction 1/(Pi+1/(Pi+1/(Pi+1/(Pi+...)))).

%C Repeat s = s + Pi; s=1/s. The initial value of s is irrelevant.

%C Solves log(x+Pi) = -log(x). This equation represents the first (by absolute value) self-intersection of the spiral defined by the polar equation r=log(theta), and this constant is the smaller value of theta in the self-intersection. - _Jeremy Tan_, Sep 03 2016

%H Michael De Vlieger, <a href="/A086773/b086773.txt">Table of n, a(n) for n = 0..10000</a>

%F Equals (sqrt(Pi^2+4)-Pi)/2 = 0.2912995... . - _R. J. Mathar_, Sep 15 2012

%e 1

%e ------

%e Pi + 1

%e ------

%e Pi + 1

%e --------

%e Pi + ...

%t RealDigits[N[(Sqrt[Pi^2 + 4] - Pi)/2, 120]] // First (* _Michael De Vlieger_, Mar 31 2015 *)

%o (PARI) default(realprecision, 2000); f(n) = s=0; for(x=1,n,s=s+Pi; s=1/s); print(s)

%Y Cf. A188722.

%K easy,nonn,cons

%O 0,1

%A _Cino Hilliard_, Aug 03 2003