A057119 -id:A057119 - cp's OEIS frontend

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.

0, 10, 1010, 101100, 10110010, 1011100100, 101100110100, 10111001001100, 1011100110100010, 101110011010011000, 10110011101001100010, 1011110010011011000100, 101100111011010001100100

Offset: 0

Antti Karttunen, Jan 27 2003

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.

			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...

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.

Compare to similar Wolframesque plots given in A122229, A122232, A122235, A122239, A122242, A122245. See also A079438, A080067, A080071, A057119.

a(n) = A007088(A080069(n)) = A063171(A080068(n)).

A057122 The binary encoding (as a rooted planar tree) of each rooted planar binary tree. See A057123 for illustration.

Original entry on oeis.org

0, 10, 180, 210, 2920, 2980, 3380, 3490, 3730, 46800, 46920, 47720, 47940, 48420, 54120, 54180, 55860, 56130, 56610, 59700, 59810, 60690, 62610, 748960, 749200, 750800, 751240, 752200, 763600, 763720, 767080, 767620, 768580, 774760, 774980

Offset: 0

Antti Karttunen, Aug 11 2000

nonn

bintree_depth_first2tree given in A057119.

Cf. A057119, A057123, A057124. Subset of A057517.

Maple
```
map(bintree_depth_first2tree, A014486);
```

cp's OEIS Frontend

A057120 Global ranks of terms of A057119.

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A057121 Local ranks of terms of A057119.

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A057122 The binary encoding (as a rooted planar tree) of each rooted planar binary tree. See A057123 for illustration.

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