A080070 - OEIS

This site is supported by donations to The OEIS Foundation.

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 (list; graph; refs; listen; history; 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	`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 A176067 A163662 * A080120 A006937 A037220` `Adjacent sequences: A080067 A080068 A080069 * A080071 A080072 A080073`
	KEYWORD	`base,nonn`
	AUTHOR	`Antti Karttunen (Firstname.Surname(AT)gmail.com) Jan 27 2003`
	EXTENSIONS	`Python program and Wolfram-like plot added by Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 14 2006.`

Content is available under The OEIS End-User License Agreement .

Last modified February 12 18:43 EST 2012. Contains 205432 sequences.