1,2

The drib tree is an infinite binary tree labeled with rational numbers. It is generated by the following iterative process: start with the rational 1; for the left subtree increment and then take the reciprocal of the current rational; for the right subtree interchange the order of the two steps: take the reciprocal and then increment. Like the Stern-Brocot and the Bird tree, the drib tree enumerates the positive rationals: A162911(n)/A162912(n).

From Yosu Yurramendi, Jul 11 2014: (Start)

If the terms (n>0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...

1,

2, 1,

3, 1, 3,2,

5, 2, 4,3,4,1, 5,3,

8, 3, 7,5,5,1, 7,4,7,3,5,4, 7,2, 8,5,

13,5,11,8,9,2,12,7,9,4,6,5,10,3,11,7,11,4,10,7,6,1,9,5,12,5,9,7,11,3,13,8,

then the sum of the m-th row is 3^m (m = 0,1,2,), and each column k is a Fibonacci sequence (a(2^(m+2)+k) = a(2^(m+1)+k) + a(2^m+k), m = 0,1,2,..., k = 0,1,2,...,2^m-1).

If the rows are written in a right-aligned fashion:

1,

2,1,

3,1, 3,2,

5,2,4,3, 4,1, 5,3,

8,3, 7,5,5,1,7,4, 7,3,5,4, 7,2, 8,5,

13,5,11,8,9,2,12,7,9,4,6,5,10,3,11,7,11,4,10,7,6,1,9,5,12,5,9,7,11,3,13,8,

then each column k also is a Fibonacci sequence.

If the sequence is considered by blocks of length 2^m, m = 0,1,2,..., the blocks of this sequence are the reverses of blocks of A162911 ( a(2^m+k) = A162911(2^(m+1)-1-k), m = 0,1,2,..., k = 0,1,2,...,2^m-1). (End)

R. Hinze, Functional pearls: the bird tree, J. Funct. Programming 19 (2009), no. 5, 491-508.

b(n) where a(1) = 1; a(2n) = b(n); a(2n+1) = a(n) + b(n); and b(1) = 1; b(2n) = a(n) + b(n); b(2n+1) = a(n).

a(2^(m+1)+2*k) = a(2^m+k) + a(2^(m+1)-1-k) , a(2^(m+1)+2*k+1) = a(2^(m+1)-1-k) , a(1) = 1 , m=0,1,2,3,... , k=0,1,...,2^m-1. - Yosu Yurramendi, Jul 11 2014

a(2^(m+1) + 2*k + 1) = A162911(2^m + k), m >= 0, 0 <= k < 2^m.

a(2^(m+1) + 2*k) = A162911(2^m + k) + a(2^m + k), m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Mar 30 2016

a(n*2^(m+1) + A096773(m)) = A268087(n), n > 0, m >= 0. - Yosu Yurramendi, Feb 20 2017

a(n) = A002487(1+A258996(n)), n > 0. - Yosu Yurramendi, Jun 23 2021

The first four levels of the drib tree:

[1/1],

[1/2, 2/1],

[2/3, 3/1, 1/3, 3/2],

[3/5, 5/2, 1/4, 4/3, 3/4, 4/1, 2/5, 5/3].

(Haskell) import Ratio; drib :: [Rational]; drib = 1 : map (recip . succ) drib \/ map (succ . recip) drib; (a : as) \/ bs = a : (bs \/ as); a162911 = map numerator drib; a162912 = map denominator drib

(R)

blocklevel <- 6 # arbitrary

a <- 1

for(m in 0:blocklevel) for(k in 0:(2^m-1)){

a[2^(m+1)+2*k] <- a[2^(m+1)-1-k] + a[2^m+k]

a[2^(m+1)+2*k+1] <- a[2^(m+1)-1-k]

}

a

# Yosu Yurramendi, Jul 11 2014

This sequence is the composition of A162910 and A059893: a(n) = A162910(A059893(n)). This sequence is a permutation of A002487(n+2).

Cf. A096773.

Sequence in context: A046073 A309786 A385200 * A230070 A224762 A039776

Adjacent sequences: A162909 A162910 A162911 * A162913 A162914 A162915

easy,frac,nonn

Ralf Hinze (ralf.hinze(AT)comlab.ox.ac.uk), Aug 05 2009

Edited by Charles R Greathouse IV, May 13 2010

approved