

A085846


Decimal expansion of root of x = (1+1/x)^x.


8



2, 2, 9, 3, 1, 6, 6, 2, 8, 7, 4, 1, 1, 8, 6, 1, 0, 3, 1, 5, 0, 8, 0, 2, 8, 2, 9, 1, 2, 5, 0, 8, 0, 5, 8, 6, 4, 3, 7, 2, 2, 5, 7, 2, 9, 0, 3, 2, 7, 1, 2, 1, 2, 4, 8, 5, 3, 7, 7, 1, 0, 3, 9, 6, 1, 6, 8, 5, 0, 6, 4, 8, 8, 0, 0, 9, 1, 5, 7, 7, 4, 3, 6, 2, 9, 0, 4, 2, 0, 1, 3, 8, 0, 4, 8, 2, 8, 2, 5, 6, 6, 1
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OFFSET

1,1


COMMENTS

Equivalently, the root of x^(x+1) = (x+1)^x.
Also a root of 1/(x^(1/x)1)  x = 0 and 1/(x^(1/x)1/x1)  x = 0, which also contains the root 5.50798565277317825758902... 1/(x^(1/x)1) ~ Pi(x) and 1/(x^(1/x)1/x1) ~ Pi(x), which is a much better approximation. These roots also can be computed by the recurrences x = 1/(x^(1/x)1) and x = 1/(x^(1/x)1/x1).  Cino Hilliard, Sep 13 2008


LINKS

Table of n, a(n) for n=1..102.
Nicolae Anghel, Foias Numbers, An. Sţiinţ. Univ. Ovidius Constanţa. Mat. (The Journal of Ovidius University of Constanţa, 2018) 26(3), 2128.
Eric Weisstein's World of Mathematics, Foias Constant


FORMULA

x satisfies x^(1/x)=(x+1)^(1/(x+1)).  Marco Matosic, Nov 25 2005


EXAMPLE

2.2931662874118610315080282912508058643722572903271212485377103961...


MATHEMATICA

RealDigits[ FindRoot[x^(1/x)  (x + 1)^(1/(x + 1)) == 0, {x, 2}, WorkingPrecision > 128][[1, 2]], 10, 111][[1]] (* Robert G. Wilson v *)


PROG

(PARI) solve(x=2, 3, (1+1/x)^xx) \\ Charles R Greathouse IV, Apr 14 2014


CROSSREFS

Cf. A021002, A169862.
Sequence in context: A308037 A011403 A113554 * A021440 A271524 A157216
Adjacent sequences: A085843 A085844 A085845 * A085847 A085848 A085849


KEYWORD

nonn,cons


AUTHOR

Eric W. Weisstein, Jul 05 2003


STATUS

approved



