OFFSET
1,3
COMMENTS
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
From Giuseppe Coppoletta, Aug 22 2016: (Start)
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.
With this we have a surprising representation of the Foias constant:
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)
Named after the Romanian-American mathematician Ciprian Foias (1933 - 2020). - Amiram Eldar, Aug 25 2020
LINKS
Giuseppe Coppoletta, Table of n, a(n) for n = 1..10000
Nicolae Anghel, Foias Numbers, An. Sţiinţ. Univ. Ovidius Constanţa. Mat. (The Journal of Ovidius University of Constanţa), Vol. 26, No. 3 (2018), pp. 21-28.
John Ewing and Ciprian Foias, An Interesting Serendipitous Real Number, in: C. Calude and G. Paun (eds.), Finite vs, Infinite: Contributions to an Eternal Dilemma, Springer, London, 2000, pp. 119-126, alternative link.
Eric Weisstein's World of Mathematics, Foias Constant.
Wikipedia, Foias Constant.
FORMULA
x_{n+1} = (1 + 1/{x_n})^n for n=1,2,3,...
EXAMPLE
1.18745235112650105459548015839651935121569268158586035301010412619878...
MATHEMATICA
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 *)
PROG
(PARI) f(x, n)=for(i=2, n, x=(1.0+1.0/x)^(i-1)); x
default(realprecision, 200);
solve(y=1, 2, f(y, 800)-1-10^(-200)) \\ Robert Gerbicz, May 08 2008
(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
(Sage) R = RealField(350); RealNumber = R; x=R(2)
for n in xsrange (220, 0, -1): x=1/(x^(1/n)-1)
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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jul 05 2003
EXTENSIONS
More terms from Robert Gerbicz, May 08 2008
More terms from Giuseppe Coppoletta, Aug 19 2016
STATUS
approved