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 A080070 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. 28
 0, 10, 1010, 101100, 10110010, 1011100100, 101100110100, 10111001001100, 1011100110100010, 101110011010011000, 10110011101001100010, 1011110010011011000100, 101100111011010001100100 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Corresponding Lisp/Scheme S-expressions are (), (()), (()()), (()(())), (()(())()), (()((())())), (()(())(()())), ... 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. LINKS Table of n, a(n) for n=0..12. A. Karttunen, Illustration of initial terms A. Karttunen, Python program for computing this sequence. A. Karttunen, Terms a(1)-a(256) plotted as a Wolframesque triangle. A. Karttunen, Terms a(1)-a(512) plotted as a Wolframesque triangle. FORMULA a(n) = A007088(A080069(n)) = A063171(A080068(n)). EXAMPLE 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: .............................................. .....\/................\/\/..........\/\/..... ......\/......\/\/......\/............\/...... .....\/........\/........\/..........\/....... ......(A057164).(A057548)..(A057163).......... ........................o..................... ........................|..................... ........o.....o.........o...o.........o....... ........|.....|..........\./..........|....... ....o...o.....o...o.......o.........o.o.o..... .....\./.......\./........|..........\|/...... ......*.........*.........*...........*....... ..[()(())]..[(())()]..[((())())]..[()(())()].. ...101100....110010....11100100....10110010... CROSSREFS Compare to similar Wolframesque plots given in A122229, A122232, A122235, A122239, A122242, A122245. See also A079438, A080067, A080071, A057119. Sequence in context: A079214 A163662 A176067 * A303610 A080120 A300571 Adjacent sequences: A080067 A080068 A080069 * A080071 A080072 A080073 KEYWORD base,nonn AUTHOR Antti Karttunen, Jan 27 2003 STATUS approved

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Last modified December 6 03:07 EST 2023. Contains 367594 sequences. (Running on oeis4.)