A162911 Numerators of drib tree fractions, where drib is the bit-reversal permutation tree of the Bird tree.

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

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 reciprocalize the current rational; for the right subtree interchange the order of the two steps: the rational is first reciprocalized and then incremented. Like the Stern-Brocot and the Bird tree, the drib tree enumerates all 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,

1, 2,

2, 3,1, 3,

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

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

...

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

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

1

1, 2

2, 3,1, 3

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

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

...

then each column k also is a Fibonacci-type 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 A162912 (a(2^m+k) = A162912(2^(m+1)-1-k), m = 0,1,2,..., k = 0..2^m-1).

(End)

From Yosu Yurramendi, Jan 12 2017: (Start)

a(2^(m+2m' ) + A020988(m')) = A000045(m+1), m>=0, m'>=0

a(2^(m+2m'+1) + A020989(m')) = A000045(m+3), m>=0, m'>=0

a(2^(m+2m' ) - 1 - A002450(m')) = A000045(m+1), m>=0, m'>=0

a(2^(m+2m'+1) - 1 - A072197(m'-1)) = A000045(m+3), m>=0, m'>0

a(2^(m+1) -1) = A000045(m+2), m>=0. (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

• Ralf 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 A162909 and A059893: a(n) = A162909(A059893(n)). This sequence is a permutation of A002487(n+1).

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

PARI

a(n) = my(x = 0, y = 1); forstep(i = logint(n, 2), 0, -1, [x, y] = if(bittest(n, i), [y, x + y], [x + y, x])); y \\ Mikhail Kurkov, Oct 12 2023

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+1)+2*k+1] <- a[2^(m+1)-1-k] + a[2^m+k]
}
a
# Yosu Yurramendi, Jul 11 2014

Formula

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

a(2^(m+1) + 2*k) = A162912(2^m + k), m >= 0, 0 <= k < 2^m.

a(2^(m+1) + 2*k + 1) = a(2^m + k) + A162912(2^m + k), m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Mar 30 2016

a(n*2^m + A176965(m)) = A268087(n), n > 0, m > 0. - Yosu Yurramendi, Feb 20 2017

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