editing

approved

Cf. A000045, A257243A257242.

approved

editing

proposed

approved

editing

proposed

The right diagonal is sequence A000045 (Fibonacci).

proposed

editing

editing

proposed

Triangle starts:

Tree starts:

1

|

1

|

2--------------

| |

1 3---------

| | |

3----- 1 5-----

| | | | |

2 4---- 4---- 2 8----

| | | | | | | |

5 1 7 3 5 7 3 13

allocated for Michel MarcusTree 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

1,3

"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."

J.-P. Allouche, K. Scheicher and R. Tichy, <a href="http://dml.cz/handle/10338.dmlcz/133301">Regular maps in generalized number systems, Math. Slovaca 50 (2000), 41-58.

B. Rittaud, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rittaud2/rittaud11.html">On the Average Growth of Random Fibonacci Sequences</a>, Journal of Integer Sequences, 10 (2007), Article 07.2.4.

1;

1;

2;

1, 3;

3, 1, 5;

2, 4, 4, 2, 8;

5, 1, 7, 3, 5, 7, 3, 13;

...

(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); ); }

Cf. A000045, A257243.

allocated

nonn,tabf

Michel Marcus, Apr 19 2015

approved

editing

allocated for Michel Marcus

allocated

approved