A199814 Decimal expansion of x value of the unique pairwise intersection on (0,1) of distinct order 5 power tower functions with parentheses inserted.

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

Offset

0,1

Comments

Order 5 is the smallest order such that pairwise intersections on (0,1) of distinct power tower functions with parentheses inserted exist. The corresponding y value is 0.66337467860163682654502... . The two intersecting functions are x-> (x^(x^x))^(x^x) and x-> x^(x^((x^x)^x)).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Vladimir Reshetnikov, Intersections of x^x^...^x, SeqFan Discussion, Nov 2011.

Eric Weisstein's World of Mathematics, Power Tower

Formula

x in (0,1) : x^(x^2)-2*x = 0.

Example

0.42801103796472992390204...

Maple

f:= x-> (x^(x^x))^(x^x): g:= x-> x^(x^((x^x)^x)):
nmax:= 140: Digits:= nmax+10:
xv:= fsolve(f(x)=g(x), x=0..0.99):
s:= convert(xv, string):
seq(parse(s[n+2]), n=0..nmax);

Mathematica

x /. FindRoot[x^(x^2) - 2*x == 0, {x, 1/2}, WorkingPrecision -> 110] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Dec 05 2013 *)

Crossrefs

Cf. A000081 (number of distinct power tower functions), A000108 (number of parenthesizations), A199879 (continued fraction), A199880 (Engel expansion).

Sequence in context: A348873 A128333 A201414 * A195347 A200693 A348681

Adjacent sequences: A199811 A199812 A199813 * A199815 A199816 A199817

Keyword

nonn,cons

Author

Alois P. Heinz, Nov 10 2011

Status

approved