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A085846 Decimal expansion of root of x = (1+1/x)^x.

%I A085846 #26 Feb 16 2025 08:32:50
%S A085846 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,
%T A085846 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,
%U A085846 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
%N A085846 Decimal expansion of root of x = (1+1/x)^x.
%C A085846 Equivalently, the root of x^(x+1) = (x+1)^x.
%C A085846 Also a root of 1/(x^(1/x)-1) - x = 0 and 1/(x^(1/x)-1/x-1) - x = 0, which also contains the root 5.50798565277317825758902... 1/(x^(1/x)-1) ~ Pi(x) and 1/(x^(1/x)-1/x-1) ~ 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/x-1). - _Cino Hilliard_, Sep 13 2008
%C A085846 This constant is transcendental (Lord, 2002). - _Amiram Eldar_, Oct 29 2022
%H A085846 Nicolae Anghel, <a href="https://doi.org/10.2478/auom-2018-0030">Foias Numbers</a>, An. Sţiinţ. Univ. Ovidius Constanţa. Mat. (The Journal of Ovidius University of Constanţa, 2018) 26(3), 21-28.
%H A085846 Nick Lord, <a href="https://www.jstor.org/stable/3621590">Two Other Transcendental Numbers Obtained by (Mis)calculating e</a>, The Mathematical Gazette, Vol. 86, No. 505 (2002), pp. 103-105.
%H A085846 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FoiasConstant.html">Foias Constant</a>.
%H A085846 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A085846 x satisfies x^(1/x) = (x+1)^(1/(x+1)). - _Marco Matosic_, Nov 25 2005
%e A085846 2.2931662874118610315080282912508058643722572903271212485377103961...
%t A085846 RealDigits[ FindRoot[x^(1/x) - (x + 1)^(1/(x + 1)) == 0, {x, 2}, WorkingPrecision -> 128][[1, 2]], 10, 111][[1]] (* _Robert G. Wilson v_ *)
%o A085846 (PARI) solve(x=2,3,(1+1/x)^x-x) \\ _Charles R Greathouse IV_, Apr 14 2014
%Y A085846 Cf. A021002, A124930, A169862.
%K A085846 nonn,cons
%O A085846 1,1
%A A085846 _Eric W. Weisstein_, Jul 05 2003