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A199814 Decimal expansion of x value of the unique pairwise intersection on (0,1) of distinct order 5 power tower functions with parentheses inserted. 3
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 (list; constant; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A050105 A128333 A201414 * A195347 A200693 A241298

Adjacent sequences:  A199811 A199812 A199813 * A199815 A199816 A199817

KEYWORD

nonn,cons

AUTHOR

Alois P. Heinz, Nov 10 2011

STATUS

approved

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Last modified February 21 07:18 EST 2018. Contains 299390 sequences. (Running on oeis4.)