A257243 Tree R defined as the subtree of A257242 tree made of all shortest walks.

1, 1, 2, 1, 3, 3, 1, 5, 2, 4, 4, 2, 8, 5, 1, 7, 3, 5, 7, 3, 13, 3, 7, 5, 3, 11, 7, 1, 9, 5, 9, 11, 5, 21, 8, 2, 12, 4, 6, 10, 4, 18, 4, 10, 6, 4, 14, 12, 2, 16, 8, 14, 18, 8, 34, 5, 11, 9, 5, 19, 9, 1, 11, 7, 13, 15, 7, 29, 11, 3, 17, 5, 7, 13, 5, 23, 7, 17

Offset

1,3

Comments

"In other words, we start from 1, with only child 1. Then, the (n-1) first rows being constructed, the n-th one is made of the nodes b such that, denoting by a their parent, the pair (a; b) did not already appear upper in the subtree (that is no row before the n-th one shows the pair(a; b)). The tree R is the restricted subtree of T."

"The sequence of labels in the tree R, read in breadth-first order is a beta-regular sequence, as defined by Allouche, Scheicher and Tichy, where here beta is the numeration system defined by the Fibonacci sequence."

The right diagonal is sequence A000045 (Fibonacci).

Links

Table of n, a(n) for n=1..78.

J.-P. Allouche, K. Scheicher and R. Tichy, Regular maps in generalized number systems, Math. Slovaca 50 (2000), 41-58.

B. Rittaud, On the Average Growth of Random Fibonacci Sequences, Journal of Integer Sequences, 10 (2007), Article 07.2.4.

Example

Triangle starts:
1;
1;
2;
1, 3;
3, 1, 5;
2, 4, 4, 2, 8;
5, 1, 7, 3, 5, 7, 3, 13;
...
Tree starts:
1
|
1
|
2--------------
|             |
1             3---------
|             |        |
3-----        1        5-----
|    |        |        |    |
2    4----    4----    2    8----
|    |   |    |   |    |    |   |
5    1   7    3   5    7    3  13

Prog

(PARI) printrow(row) = for (k=1, #row, if (row[k]>0, print1(row[k], ", "))); print();
dchild(a, b) = b-a;
schild(a, b) = b+a;
tablr(nn) = {printrow(prow = [1]); printrow(crow = [1]); nrow = vector(2); nrow[2] = schild(prow[1], crow[1]); printrow(nrow); for (n=4, nn, prow = crow; crow = nrow; nrow = vector(4*#prow); inew = 0; ichild = 0; for (inode=1, #prow, node = prow[inode]; child = crow[ichild++]; if (child > 0, nrow[inew++] = dchild(node, child); nrow[inew++] = schild(node, child), nrow[inew++] = -1; nrow[inew++] = -1); child = crow[ichild++]; if (child > 0, nrow[inew++] = dchild(node, child); nrow[inew++] = schild(node, child), nrow[inew++] = -1; nrow[inew++] = -1); ); printrow(nrow); ); }

Crossrefs

Cf. A000045, A257242.

Sequence in context: A376479 A142249 A274705 * A097351 A207330 A048600

Adjacent sequences: A257240 A257241 A257242 * A257244 A257245 A257246

Keyword

nonn,tabf

Author

Michel Marcus, Apr 19 2015

Status

approved