0,2

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.

A. Karttunen, Illustration of initial terms

A. Karttunen, Python program for computing this sequence.

A. Karttunen, Terms a(1)-a(512) plotted as a Wolframesque triangle.

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

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...

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

Sequence in context: A075171 A106456 A079214 this_sequence A080120 A006937 A037220

Adjacent sequences: A080067 A080068 A080069 this_sequence A080071 A080072 A080073

base,nonn

Antti Karttunen (Firstname.Surname(AT)gmail.com) Jan 27 2003

Python program and Wolfram-like plot added by Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 14 2006.