

A085848


Decimal expansion of Foias' Constant.


1



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, 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, 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
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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/(Nn))1) for n = 1..N1, 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)


LINKS

Giuseppe Coppoletta, Table of n, a(n) for n = 1..10000
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[n1, a])^(n1); RealDigits[ a /. FindRoot[x[220, a] == 10^65, {a, 1, 2}, WorkingPrecision > 110, MaxIterations > 500]][[1]][[1 ;; 105]] (* JeanFrançois Alcover, Nov 12 2012 *)


PROG

(PARI) f(x, n)=for(i=2, n, x=(1.0+1.0/x)^(i1)); x
default(realprecision, 200);
solve(y=1, 2, f(y, 800)110^(200)) \\ Robert Gerbicz, May 08 2008
(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

Sequence in context: A274442 A249136 A154815 * A008960 A077744 A111448
Adjacent sequences: A085845 A085846 A085847 * A085849 A085850 A085851


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



