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%I #54 Dec 04 2021 12:49:26
%S 1,1,8,7,4,5,2,3,5,1,1,2,6,5,0,1,0,5,4,5,9,5,4,8,0,1,5,8,3,9,6,5,1,9,
%T 3,5,1,2,1,5,6,9,2,6,8,1,5,8,5,8,6,0,3,5,3,0,1,0,1,0,4,1,2,6,1,9,8,7,
%U 8,0,4,1,8,7,2,3,5,2,5,4,0,7,3,8,7,0,2,4,6,5,7,6,0,6,0,8,6,5,7,9,4,3,3,7,8
%N Decimal expansion of Foias' constant.
%C This is the unique real x_1 such that iterating x_{n+1} = (1 + 1/x_n)^n yields a series which diverges to infinity (rather than having 1 and infinity as limit points). - _Charles R Greathouse IV_, Nov 19 2013
%C From _Giuseppe Coppoletta_, Aug 22 2016: (Start)
%C It appears that x_1 can be easily backward calculated. Let us define for any fixed N, t_(n+1) = 1/((t_n)^(1/(N-n))-1) for n = 1..N-1, beginning with whatever t_1 > 1. Then t_N approaches x_1 as N tends to infinity. If we allow t_n to be complex, this is still true for any t_1 in the complex domain, excluding t_1 = 1.
%C With this we have a surprising representation of the Foias constant:
%C x_1 = 1/(-1+1/(-1+exp(-1/2*log(abs(-1+exp(-1/3*log(abs(-1+exp(-1/4*log(abs(-1+exp(-1/5*... (End)
%C Named after the Romanian-American mathematician Ciprian Foias (1933 - 2020). - _Amiram Eldar_, Aug 25 2020
%H Giuseppe Coppoletta, <a href="/A085848/b085848.txt">Table of n, a(n) for n = 1..10000</a>
%H Nicolae Anghel, <a href="https://doi.org/10.2478/auom-2018-0030">Foias Numbers</a>, An. Sţiinţ. Univ. Ovidius Constanţa. Mat. (The Journal of Ovidius University of Constanţa), Vol. 26, No. 3 (2018), pp. 21-28.
%H John Ewing and Ciprian Foias, <a href="https://doi.org/10.1007/978-1-4471-0751-4_8">An Interesting Serendipitous Real Number</a>, in: C. Calude and G. Paun (eds.), Finite vs, Infinite: Contributions to an Eternal Dilemma, Springer, London, 2000, pp. 119-126, <a href="https://www.cs.auckland.ac.nz/~cristian/finitevsinfinite.pdf">alternative link</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FoiasConstant.html">Foias Constant</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Foias_constant">Foias Constant</a>.
%F x_{n+1} = (1 + 1/{x_n})^n for n=1,2,3,...
%e 1.18745235112650105459548015839651935121569268158586035301010412619878...
%t x[1, a_] = a; x[n_, a_] :=(1+1/x[n-1, a])^(n-1); RealDigits[ a /. FindRoot[x[220, a] == 10^65, {a, 1, 2}, WorkingPrecision -> 110, MaxIterations -> 500]][[1]][[1 ;; 105]] (* _Jean-François Alcover_, Nov 12 2012 *)
%o (PARI) f(x,n)=for(i=2,n,x=(1.0+1.0/x)^(i-1));x
%o default(realprecision,200);
%o solve(y=1,2,f(y,800)-1-10^(-200)) \\ _Robert Gerbicz_, May 08 2008
%o (PARI) foias(p)=my(N=2*p,t=2);localprec(p);for(n=1,N-1,t=1./(t^(1/(N-n))-1));t \\ returns the Foias constant to p decimals; Bill Allombert, via _Charles R Greathouse IV_, Jan 04 2017
%o (Sage) R = RealField(350); RealNumber = R; x=R(2)
%o for n in xsrange (220,0,-1): x=1/(x^(1/n)-1)
%o print('x_1 =',x); print('digits x_1 =',[ZZ(k) for k in x.str(skip_zeroes=True) if k.isdigit()]) # _Giuseppe Coppoletta_, Aug 22 2016
%K nonn,cons
%O 1,3
%A _Eric W. Weisstein_, Jul 05 2003
%E More terms from _Robert Gerbicz_, May 08 2008
%E More terms from _Giuseppe Coppoletta_, Aug 19 2016
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