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%I M1179 #35 Jan 31 2022 01:12:39
%S 1,1,2,4,9,20,48,114,282,703,1787,4583,11900,31131,82117,217954,
%T 581970,1561704,4210263,11396488,30963024,84402984,230779071,
%U 632762424,1739387089
%N Number of distinct values taken by 4^4^...^4 (with n 4's and parentheses inserted in all possible ways).
%C See also the Four Fours puzzle [Bourke]. Four fours is a mathematical puzzle. The goal of four fours is to find the simplest mathematical expression for every whole number from 0 to some maximum, using only common mathematical symbols and the digit four (no other digit is allowed). The subsequence of primes begins 2, 1787, 4583, no more through a(23). [_Jonathan Vos Post_, Apr 02 2011]
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. K. Guy and J. L. Selfridge, <a href="http://www.jstor.org/stable/2319392">The nesting and roosting habits of the laddered parenthesis</a>, Amer. Math. Monthly 80 (8) (1973), 868-876.
%H Paul Bourke, <a href="http://paulbourke.net/fun/4444/">Four Fours Problem</a>.
%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>
%H <a href="http://mathoverflow.net/questions/79442/number-of-distinct-values-taken-by-xx-x-with-parentheses-inserted-in-all-pos">MathOverflow discussion of related questions</a>
%Y Cf. A002845, A003018, A145545, A145546, A145547, A145548, A145549, A145550, A000081.
%K nonn,nice,more
%O 1,3
%A _N. J. A. Sloane_
%E a(12)-a(23) from _Jon E. Schoenfield_, Oct 11 2008
%E a(24)-a(25) from _Marek Hubal_, Mar 01 2019
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