|
%I M1139 N0435 #74 Apr 21 2024 14:01:08
%S 1,1,1,2,4,8,17,36,78,171,379,851,1928,4396,10087,23273,53948,125608,
%T 293543,688366,1619087,3818818,9029719,21400706,50828664,120963298,
%U 288405081,688821573,1647853491,3948189131,9473431479
%N Number of distinct values taken by 2^2^...^2 (with n 2's and parentheses inserted in all possible ways).
%C a(n) <= A002955(n). - _Max Alekseyev_, Sep 23 2009
%D J. Q. Longyear, personal communication.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H F. Goebel and R. P. Nederpelt, <a href="http://www.jstor.org/stable/2316312">The number of numerical outcomes of iterated powers</a>, Amer. Math. Monthly, 80 (1971), 1097-1103.
%H R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: <a href="/A002572/a002572.jpg">front</a>, <a href="/A002572/a002572_1.jpg">back</a> [Annotated scanned copy, with permission]
%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 R. K. Guy and J. L. Selfridge, <a href="/A003018/a003018.pdf">The nesting and roosting habits of the laddered parenthesis</a> (annotated cached copy, with permission)
%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a002/A002845.java">Java program</a> (github).
%H J. Longyear, T. Trotter, and N. J. A. Sloane, <a href="/A005321/a005321_2.pdf">Correspondence</a>.
%H MathOverflow, <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>.
%H Vladimir Reshetnikov, <a href="https://github.com/VladimirReshetnikov/Oeis.A002845">Computes terms of OEIS sequence A002845 (C# library)</a>.
%H Jon E. Schoenfield, <a href="/A002845/a002845.txt">The 851 values for n=12</a>.
%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>
%e From _M. F. Hasler_, Apr 17 2024: (Start)
%e The table with explicit lists of values starts as follows:
%e n | distinct values of 2^...^2 with all possible parenthesizations
%e -----+---------------------------------------------------------------
%e 1 | 2
%e 2 | 2^2 = 4
%e 3 | (2^2)^2 = 2^(2^2) = 16
%e 4 | (2^2^2)^2 = 2^8 = 256, (2^2)^(2^2) = 2^(2^2^2) = 2^16 (= 65536)
%e 5 | 256^2 = 2^16, (2^16)^2 = 2^32, 2^256, 2^2^16 (~ 2*10^19728)
%e 6 | (2^16)^2 = 2^32, 2^64, 2^512, 2^2^16, 2^2^17, 2^2^32, 2^2^256, 2^2^2^16
%e 7 | 2^64, 2^128, 2^256, 2^1024, 2^2^17, 2^2^18, 2^2^32, 2^2^33, 2^2^64, 2^2^257,
%e | 2^2^512, 2^2^2^16, 2^2^65537, 2^2^2^17, 2^2^2^32, 2^2^2^256, 2^2^2^2^16
%e ...| ...
%e (When parentheses are omitted above, we use that ^ is right associative.) (End)
%Y Cf. A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A000081.
%K nonn,nice,more,changed
%O 1,4
%A _N. J. A. Sloane_
%E a(12)-a(13) corrected and a(14)-a(27) added by _Jon E. Schoenfield_, Oct 11 2008
%E a(28)-a(29) computed by Kirill Osenkov, added by _Vladimir Reshetnikov_, Feb 07 2019
%E a(30)-a(31) added by _Sean A. Irvine_, Feb 18 2019
|
