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A071153 Łukasiewicz word for each rooted plane tree (interpretation e in Stanley's exercise 19) encoded by A014486 (or A063171), with the last leaf implicit, i.e., these words are given without the last trailing zero, except for the null tree which is encoded as 0. 13
0, 1, 20, 11, 300, 201, 210, 120, 111, 4000, 3001, 3010, 2020, 2011, 3100, 2101, 2200, 1300, 1201, 2110, 1210, 1120, 1111, 50000, 40001, 40010, 30020, 30011, 40100, 30101, 30200, 20300, 20201, 30110, 20210, 20120, 20111, 41000, 31001, 31010 (list; graph; refs; listen; history; text; internal format)



Note: this finite decimal representation works only up to the 6917th term, as the 6918th such word is already (10,0,0,0,0,0,0,0,0,0). The sequence A071154 shows the initial portion of this sequence sorted.


Table of n, a(n) for n=0..39.

A. Karttunen, Gatomorphisms and other excursions amidst the plane trees and parenthesizations (Includes the complete Scheme program for computing this sequence)

R. P. Stanley, Hipparchus, Plutarch, Schröder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.

R. P. Stanley, Exercises on Catalan and Related Numbers

OEIS Wiki, Łukasiewicz words

Index entries for sequences related to Łukasiewicz

Index entries for sequences related to parenthesizing


The 11th term of A063171 is 10110010, corresponding to parenthesization ()(())(), thus its Łukasiewicz word is 3010. The 18th term of A063171 is 11011000, corresponding to parenthesization (()(())), thus its Łukasiewicz word is 1201. I.e., in the latter example there is one list on the top-level, which in turn contains two sublists, of which the first is zero elements long and the second is a sublist containing one empty sublist (the last zero is omitted).


For n >= 1, the number of zeros in the term a(n) is given by A057514(n)-1.

The first digit of each term is given by A057515.

Cf. A014486, A059984, A059985, A071152, A071154.

Corresponding factorial walk encoding: A071155 (A071157, A071159).

a(n) = A079436(n)/10.

Sequence in context: A298208 A247337 A071160 * A209644 A154043 A073868

Adjacent sequences:  A071150 A071151 A071152 * A071154 A071155 A071156




Antti Karttunen, May 14 2002



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Last modified November 20 02:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)