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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 (list; constant; graph; refs; listen; history; text; internal format)
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)

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[n-1, a])^(n-1); RealDigits[ a /. FindRoot[x[220, a] == 10^65, {a, 1, 2}, WorkingPrecision -> 110, MaxIterations -> 500]][[1]][[1 ;; 105]] (* Jean-Fran├žois 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

(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

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Last modified December 9 00:45 EST 2016. Contains 278959 sequences.