A257243 - OEIS

%I #11 Jan 08 2025 10:58:50

%S 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,

%T 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,

%U 1,11,7,13,15,7,29,11,3,17,5,7,13,5,23,7,17

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

%C "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."

%C "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."

%C The right diagonal is sequence A000045 (Fibonacci).

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

%H 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.

%e Triangle starts:

%e 1;

%e 1;

%e 2;

%e 1, 3;

%e 3, 1, 5;

%e 2, 4, 4, 2, 8;

%e 5, 1, 7, 3, 5, 7, 3, 13;

%e ...

%e Tree starts:

%e 1

%e |

%e 1

%e |

%e 2--------------

%e |             |

%e 1             3---------

%e |             |        |

%e 3-----        1        5-----

%e |    |        |        |    |

%e 2    4----    4----    2    8----

%e |    |   |    |   |    |    |   |

%e 5    1   7    3   5    7    3  13

%o (PARI) printrow(row) = for (k=1, #row, if (row[k]>0, print1(row[k], ", "))); print();

%o dchild(a,b) = b-a;

%o schild(a,b) = b+a;

%o 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););}

%Y Cf. A000045, A257242.

%K nonn,tabf

%O 1,3

%A _Michel Marcus_, Apr 19 2015