A194346 Decimal expansion of h_o(1/17), where h_o(x) is the odd infinite power tower function.

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, 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, 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

Offset: 0

Author

Jonathan Sondow, Aug 27 2011

Keywords

• nonn
• cons

Comments

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.

h_o(1/17) and h_e(1/17) are irrational, and at least one of them is transcendental (see Sondow and Marques 2010).

Examples

			0.204274736665518499175698745186446957991668690348422572736592466759324966133336...
		

References

• See the References in Sondow and Marques 2010.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see Definition 4.3, Figure 7, and top of p. 163.

Crossrefs

Cf. A073229, A073230, A073243, A194347.

Programs

Mathematica

a = N[1/17, 100]; Do[a = (1/17)^(1/17)^a, {3000}]; RealDigits[a, 10, 100] // First
RealDigits[ Fold[ N[#2^#1, 128] &, 1/17, Table[1/17, {5710}]], 10, 105][[1]] (* Robert G. Wilson v, Mar 20 2012 *)

PARI

solve(x=0,1,17^(-17^-x)-x) \\ Charles R Greathouse IV, Mar 20 2012