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A003018 Number of distinct values taken by 3^3^...^3 (with n 3's and parentheses inserted in all possible ways).
(Formerly M1178)
1, 1, 2, 4, 9, 20, 47, 111, 270, 664, 1659, 4184, 10662, 27367, 70747, 183925, 480656, 1261630, 3324772, 8792592, 23327249, 62067785, 165586565 (list; graph; refs; listen; history; text; internal format)



The g.f. (-1+z+6*z**2-3*z**3-8*z**4+3*z**5)/(-1+2*z+6*z**2-9*z**3-10*z**4+9*z**5) conjectured by Simon Plouffe in his 1992 dissertation is wrong.


F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis. Amer. Math. Monthly 80 (1973), 868-876.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


Table of n, a(n) for n=1..23.

R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy)

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for sequences related to parenthesizing

MathOverflow discussion of related questions


Cf. A002845, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A000081.

Sequence in context: A130802 A022543 A036618 * A196244 A035084 A213905

Adjacent sequences:  A003015 A003016 A003017 * A003019 A003020 A003021




N. J. A. Sloane.


a(12) - a(23) from Jon E. Schoenfield, Oct 11 2008



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Last modified December 22 14:10 EST 2014. Contains 252364 sequences.