A162910 Denominators of Bird tree fractions.

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

Offset

1,2

Comments

The Bird tree is an infinite binary tree labeled with rational numbers. The root is labeled with 1. The tree enjoys the following fractal property: it can be transformed into its left subtree by first incrementing and then reciprocalizing the elements; for the right subtree interchange the order of the two steps: the elements are first reciprocalized and then incremented. Like the Stern-Brocot tree, the Bird tree enumerates all the positive rationals (A162909(n)/A162910(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, 3,1, 2,

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

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

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

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

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

1,

2,1,

3,3,1,2,

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

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

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

then each column k also is a Fibonacci sequence.

The Fibonacci sequences of both triangles are equal except the first terms of second triangle.

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 A162909 ( a(2^m+k) = A162909(2^(m+1)-1-k), m = 0,1,2,..., k = 0,1,2,...,2^m-1).

(End)

Links

Table of n, a(n) for n=1..88.

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

Index entries for fraction trees

Formula

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

If k is odd a(A080675(n)*2^m+k) = A268087(2^m+k), if k is even a(A136412(2^m+k+1)*2^m+k) = A268087(2^m+k), m >= 0, 0 <= k < 2^m, n > 0. a(A081254(n)) = 1, n > 0. - Yosu Yurramendi, Feb 21 2017

a(n) = A002487(1+A258996(A059893(n))) = A002487(1+A059893(A258746(n))), n > 0. - Yosu Yurramendi, Jul 14 2021

Example

The first four levels of the Bird tree: [1/1] [1/2, 2/1] [2/3, 1/3, 3/1, 3/2], [3/5, 3/4, 1/4, 2/5, 5/2, 4/1, 4/3, 5/3].

Prog

(Haskell)
import Ratio; bird :: [Rational]; bird = branch (recip . succ) (succ . recip) 1; branch f g a = a : branch f g (f a) \/ branch f g (g a); (a : as) \/ bs = a : (bs \/ as); a162909 = map numerator bird; a162910 = map denominator bird
(R)
blocklevel <- 6 # arbitrary
a <- 1
for(m in 1:blocklevel) for(k in 0:(2^(m-1)-1)){
a[2^m+k]         = a[2^m-k-1] + a[2^(m-1)+k]
a[2^m+2^(m-1)+k] = a[2^m-k-1]
}
a
# Yosu Yurramendi, Jul 11 2014

Crossrefs

This sequence is the composition of A162912 and A059893: a(n) = A162912(A059893(n)). This sequence is a permutation of A002487(n+2).

Sequence in context: A352356 A229742 A126572 * A098975 A309213 A377433

Adjacent sequences: A162907 A162908 A162909 * A162911 A162912 A162913

Keyword

easy,frac,nonn

Author

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

Status

approved