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A020651 Denominators in recursive bijection from positive integers to positive rationals (the bijection is f(1) = 1, f(2n) = f(n)+1, f(2n+1) = 1/(f(n)+1)). 7
1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 4, 2, 5, 3, 5, 1, 5, 4, 5, 3, 7, 4, 7, 2, 7, 5, 7, 3, 8, 5, 8, 1, 6, 5, 6, 4, 9, 5, 9, 3, 10, 7, 10, 4, 11, 7, 11, 2, 9, 7, 9, 5, 12, 7, 12, 3, 11, 8, 11, 5, 13, 8, 13, 1, 7, 6, 7, 5, 11, 6, 11, 4, 13, 9, 13, 5, 14, 9, 14, 3, 13, 10, 13, 7, 17, 10, 17, 4, 15, 11, 15, 7, 18, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Numerators in left-hand half of Kepler's tree of fractions. Form a tree of fractions by beginning with 1/1 and then giving every node i/j two descendants labeled i/(i+j) and j/(i+j). See A086592 for denominators.

Level n of the tree consists of 2^n nodes: 1/2; 1/3, 2/3; 1/4, 3/4, 2/5, 3/5; 1 /5, 4/5, 3/7, 4/7, 2/7, 5/7, 3/8, 5/8; ... Fibonacci numbers occur at the right edge this tree, i.e., a(A000225(n)) = A000045(n+1). The fractions are given in their reduced form, thus gcd(A020650(n), A020651(n)) = 1 and gcd(A020651(n), A086592(n)) = 1 for all n. - Antti Karttunen, May 26 2004

A generalization which includes the "rabbit tree" (A226080) and "all rationals tree" (A226130) follows.  Suppose that a,b,c,d,e,f,g,h are complex numbers.  Let S be the set of numbers defined by these rules:  (1) 1 is in S; (2) if x is in S and cx+d is not 0, then U(x) = (ax+b)/(cx+d) is in S; (3) if x is in S and gx+h is not 0, then D(x) = (ex+f)/(gx+h) is in S.  If an infinite path in the resulting tree has convergent nodes, then there is some node after which the path is "updown zigzag" ((UoD)o(UoD)o ...)  or "downup zigzag" (DoU)o(DoU)o ...).  If ag+ch is not 0, then the updown zigzag limit is invariant of x and equals [ae + cf - bg - dh + sqrt(X)]/(2(ag + ch)), where X = (ae + cf - bg - dh)^2 + 4(be + df + ag + ch).  If ce + dg is not 0, then the downup zigzag limit is invariant of x and equals [ae + bg - cf - dh + sqrt(Y)]/(2(ce + dg)), where Y = (ae + bg - cf - dh)^2 + 4(af + bh)(ce + dg)) = X.  Thus, for the tree A020651, the updown zigzag limit is -1 + sqrt(2) and the downup zigzag limit, sqrt(2). - Clark Kimberling, Nov 10 2013

From Yosu Yurramendi, Jul 13 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,

1,3,2,3,

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

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

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

then the sum of the m-th row is 3^m (m = 0,1,2,), and each column is an arithmetic sequence. The differences of the arithmetic sequences, except the first on the left, give the sequence A093873 (Numerators in Kepler's tree of harmonic fractions) (a(2^(m+1)-1-k) - a(2^m-1-k) = A093873(k), m = 0,1,2,..., k = 0,1,2,...,2^m-1).

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

                                                                        1,

                                                                     1, 2,

                                                                1, 3,2, 3,

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

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

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

then each column k is a Fibonacci sequence.

(End)

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

Johannes Kepler, Excerpt from the Chapter II of the Book III of the Harmony of the World: On the seven harmonic divisions of the string (illustrates the A020651/A086592-tree).

FORMULA

a(1) = 1, a(2n) = a(n), a(2n+1) = A020650(2n). - Antti Karttunen, May 26 2004

a(2n) = A020650(2n+1). - Yosu Yurramendi, Jul 17 2014

EXAMPLE

1, 2, 1/2, 3, 1/3, 3/2, 2/3, 4, 1/4, 4/3, ...

MAPLE

A020651 := n -> `if`((n < 2), n, `if`(type(n, even), A020651(n/2), A020650(n-1)));

MATHEMATICA

f[1] = 1; f[n_?EvenQ] := f[n] = f[n/2]+1; f[n_?OddQ] := f[n] = 1/(f[(n-1)/2]+1); a[n_] := Denominator[f[n]]; Table[a[n], {n, 1, 94}] (* Jean-Fran├žois Alcover, Nov 22 2011 *)

PROG

(Haskell)

import Data.List (transpose); import Data.Ratio (denominator)

a020651_list = map denominator ks where

   ks = 1 : concat (transpose [map (+ 1) ks, map (recip . (+ 1)) ks])

-- Reinhard Zumkeller, Feb 22 2014

(R)

N <- 25 # arbitrary

a <- c(1, 2, 1)

for(n in 1:N){

  a[4*n]   <- a[2*n]

  a[4*n+1] <- a[2*n] + a[2*n+1]

  a[4*n+2] <-          a[2*n+1]

  a[4*n+3] <- a[2*n] + a[2*n+1]

}

a

# Yosu Yurramendi, Jul 13 2014

CROSSREFS

See A093873/A093875 for the full Kepler tree.

Cf. A020650, A086592, A093873.

Sequence in context: A038568 A071912 A070940 * A002487 A245328 A060162

Adjacent sequences:  A020648 A020649 A020650 * A020652 A020653 A020654

KEYWORD

nonn,easy,frac,nice,changed

AUTHOR

David W. Wilson

EXTENSIONS

Entry revised by N. J. A. Sloane, May 24 2004

STATUS

approved

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Last modified July 28 12:23 EDT 2014. Contains 244997 sequences.