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A085848
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Decimal expansion of Foias' constant.
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2
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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
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1,3
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COMMENTS
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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
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
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LINKS
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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.
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FORMULA
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x_{n+1} = (1 + 1/{x_n})^n for n=1,2,3,...
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EXAMPLE
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1.18745235112650105459548015839651935121569268158586035301010412619878...
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MATHEMATICA
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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 *)
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PROG
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(PARI) f(x, n)=for(i=2, n, x=(1.0+1.0/x)^(i-1)); x
default(realprecision, 200);
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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