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A085184 Sequence A085183 shown in base 4. Quaternary code for binary trees. 8
0, 1, 2, 11, 12, 21, 22, 30, 111, 112, 121, 122, 130, 211, 212, 221, 222, 230, 301, 302, 310, 320, 1111, 1112, 1121, 1122, 1130, 1211, 1212, 1221, 1222, 1230, 1301, 1302, 1310, 1320, 2111, 2112, 2121, 2122, 2130, 2211, 2212, 2221, 2222, 2230, 2301, 2302 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This sequence gives two alternative ways to encode rooted plane binary trees (Stanley's interpretation 'c' = interpretation 'd' without the outermost edges):

A: scan each term from left to right and for each 0, add a leaf node to the tree (terminate a branch), for each 1, add a leftward leaning branch \, for each 2, add a rightward leaning branch / and for each 3, add a double-branch \/ and continue in left-to-right, depth-first fashion.

B: Like method A, but the roles of digits 1 and 2 are swapped. When one compares the generated trees to the "standard order" as specified in the illustrations for A014486, one obtains the permutation A074684/A074683 for the case A and A082356/A082355 for the case B.

If we assign the following weights for each digit: w(0) = -1, w(1) = w(2) = 0, w(3) = +1, then the sequence gives all base-4 numbers for which all the partial sums of digit weights (from the most significant to the least significant end) are nonnegative and the final sum is zero. The initial term 0 is considered to have no significant digits at all, so its total weight is zero also.

LINKS

Table of n, a(n) for n=1..48.

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

FORMULA

a(n) = A007090(A085183(n))

EXAMPLE

For the eleven first terms the following binary trees are constructed with method A. With method B we would get their mirror images, although it doesn't hold in general (e.g. for terms like 301-320).

........................................................\......./......\...

.....................\......./.......\......./...........\......\....../...

..*......\....../.....\......\......./....../.....\/......\......\.....\...

..0......1......2.....11.....12.....21.....22.....30....111....112....121..

CROSSREFS

Cf. A085185. Number of terms with n significant digits is given by A000108(n+1).

Sequence in context: A136986 A096109 A136997 * A037089 A038118 A038117

Adjacent sequences:  A085181 A085182 A085183 * A085185 A085186 A085187

KEYWORD

nonn,base

AUTHOR

Antti Karttunen Jun 14 2003

STATUS

approved

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Last modified May 22 13:18 EDT 2017. Contains 286872 sequences.