A162912 Denominators of drib tree fractions, where drib is the bit-reversal permutation tree of the Bird tree.

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

Offset: 1

Author

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

Keywords

• easy
• frac
• nonn

Comments

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)

Examples

			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].
		

Links

• R. Hinze, Functional pearls: the bird tree, J. Funct. Programming 19 (2009), no. 5, 491-508.
• Index entries for fraction trees

Crossrefs

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.

Programs

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

Formula

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

Extensions

Edited by Charles R Greathouse IV, May 13 2010