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%I #17 Apr 28 2020 11:58:50
%S 0,10,1010,101100,10110010,1011100100,101100110100,10111001001100,
%T 1011100110100010,101110011010011000,10110011101001100010,
%U 1011110010011011000100,101100111011010001100100
%N Decimal encoding of parenthesizations produced by simple iteration starting from empty parentheses and where each successive parenthesization is obtained from the previous by reflecting it as a general tree/parenthesization, then adding an extra stem below the root and then reflecting the underlying binary tree.
%C Corresponding Lisp/Scheme S-expressions are (), (()), (()()), (()(())), (()(())()), (()((())())), (()(())(()())), ...
%C Conjecture: only the terms in positions 0,1,2 and 4 are symmetric, i.e., A057164(A080068(n)) = A080068(n) (equivalently: A036044(A080069(n)) = A080069(n)) only when n is one of {0,1,2,4}. If this is true, then the formula given in A079438 is exact. I (AK) have checked this up to n=404631 with no other occurrence of a symmetric (general) tree.
%H A. Karttunen, <a href="/A080070/a080070.pdf">Illustration of initial terms</a>
%H A. Karttunen, <a href="/A080069/a080069.py.txt">Python program for computing this sequence.</a>
%H A. Karttunen, <a href="/A080069/a080069_256.png">Terms a(1)-a(256) plotted as a Wolframesque triangle.</a>
%H A. Karttunen, <a href="/A080069/a080069_512.png">Terms a(1)-a(512) plotted as a Wolframesque triangle.</a>
%F a(n) = A007088(A080069(n)) = A063171(A080068(n)).
%e This demonstrates how to get the fourth term 10110010 from the 3rd term 101100. The corresponding binary and general trees plus parenthesizations are shown. The first operation reflects the general tree, the second adds a new stem under the root and the third reflects the underlying binary tree, which induces changes on the corresponding general tree:
%e ..............................................
%e .....\/................\/\/..........\/\/.....
%e ......\/......\/\/......\/............\/......
%e .....\/........\/........\/..........\/.......
%e ......(A057164).(A057548)..(A057163)..........
%e ........................o.....................
%e ........................|.....................
%e ........o.....o.........o...o.........o.......
%e ........|.....|..........\./..........|.......
%e ....o...o.....o...o.......o.........o.o.o.....
%e .....\./.......\./........|..........\|/......
%e ......*.........*.........*...........*.......
%e ..[()(())]..[(())()]..[((())())]..[()(())()]..
%e ...101100....110010....11100100....10110010...
%Y Compare to similar Wolframesque plots given in A122229, A122232, A122235, A122239, A122242, A122245. See also A079438, A080067, A080071, A057119.
%K base,nonn
%O 0,2
%A _Antti Karttunen_, Jan 27 2003
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