A085848 Decimal expansion of Foias' constant.

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

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

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.4, p. 430.

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

Sequence in context: A274442 A249136 A154815 * A008960 A077744 A111448

Adjacent sequences: A085845 A085846 A085847 * A085849 A085850 A085851

Keyword

nonn,cons,changed

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