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%I #23 Feb 28 2022 12:19:36
%S 2,0,4,2,7,4,7,3,6,6,6,5,5,1,8,4,9,9,1,7,5,6,9,8,7,4,5,1,8,6,4,4,6,9,
%T 5,7,9,9,1,6,6,8,6,9,0,3,4,8,4,2,2,5,7,2,7,3,6,5,9,2,4,6,6,7,5,9,3,2,
%U 4,9,6,6,1,3,3,3,3,6,6,8,4,1,4,3,5,8,7,7,1,6,3,7,2,0,1,9,7,4,6,3
%N Decimal expansion of h_o(1/17), where h_o(x) is the odd infinite power tower function.
%C The odd infinite power tower function is h_o(x) = lim f(n,x) as n --> infinity, where f(n+1,x) = x^x^(f(n,x)) and f(1,x) = x. The even infinite power tower function h_e(x) is the same limit except with f(1,x) = x^x (see A194347). The limits exist if and only if 0 < x <= e^(1/e). If (1/e)^e <= x <= e^(1/e), then h_o(x) = h_e(x) = h(x) (the infinite power tower function-see the comments in A073230) and y = h(x) is a solution of x^y = y. If 0 < x < (1/e)^e, then h_o(x) < h_e(x), and two solutions of x^x^y = y are y = h_o(x) and y = h_e(x). For example, y = h_o(1/16) = 1/4 and y = h_e(1/16) = 1/2 are solutions of (1/16)^(1/16)^y = y.
%C h_o(1/17) and h_e(1/17) are irrational, and at least one of them is transcendental (see Sondow and Marques 2010).
%D See the References in Sondow and Marques 2010.
%H G. C. Greubel, <a href="/A194346/b194346.txt">Table of n, a(n) for n = 0..10000</a>
%H J. Sondow and D. Marques, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_37_from151to164.pdf">Algebraic and transcendental solutions of some exponential equations</a>, Annales Mathematicae et Informaticae 37 (2010) 151-164; see Definition 4.3, Figure 7, and top of p. 163.
%e 0.204274736665518499175698745186446957991668690348422572736592466759324966133336...
%t a = N[1/17, 100]; Do[a = (1/17)^(1/17)^a, {3000}]; RealDigits[a, 10, 100] // First
%t RealDigits[ Fold[ N[#2^#1, 128] &, 1/17, Table[1/17, {5710}]], 10, 105][[1]] (* _Robert G. Wilson v_, Mar 20 2012 *)
%o (PARI) solve(x=0,1,17^(-17^-x)-x) \\ _Charles R Greathouse IV_, Mar 20 2012
%Y Cf. A073229, A073230, A073243, A194347.
%K nonn,cons
%O 0,1
%A _Jonathan Sondow_, Aug 27 2011
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