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A093875 Denominators in Kepler's tree of harmonic fractions. 8
1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 4, 4, 5, 5, 5, 5, 7, 7, 7, 7, 8, 8, 5, 5, 7, 7, 7, 7, 8, 8, 6, 6, 9, 9, 10, 10, 11, 11, 9, 9, 12, 12, 11, 11, 13, 13, 6, 6, 9, 9, 10, 10, 11, 11, 9, 9, 12, 12, 11, 11, 13, 13, 7, 7, 11, 11, 13, 13, 14, 14, 13, 13, 17, 17, 15, 15, 18, 18, 11, 11, 16, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

It appears that A071585 is a bisection of this sequence, which itself is a bisection of A093873. - Yosu Yurramendi, Jan 09 2016

LINKS

R. Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for fraction trees

FORMULA

a(n) = a([n/2]) + A093873([n/2]).

Conjecture of the comment in detail: a(2n+1) = a(2n), n > 0;  a(2n+1) = A071585(n), n >= 0; a(2n) = A071585(n), n > 0. - Yosu Yurramendi, Jun 22 2016

a(2n) - A093873(2n) = a(n), n > 0; a(2n+1) - A093873(2n+1) = A093873(n), n > 0. - Yosu Yurramendi, Jul 23 2016

From Yosu Yurramendi, Jul 25 2016: (Start)

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

a(2^(m+1) + k) - a(2^m + k) = a(k),   m > 0, 0 <= k < 2^m. For k=0, a(0) = 1 is needed.

a(2^(m+2) - k - 1) = a(2^(m+1) - k - 1) + a(2^m - k - 1), m >= 0, 0 <= k < 2^m. (End)

EXAMPLE

The first few fractions are:

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

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ...

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

MATHEMATICA

num[1] = num[2] = 1; den[1] = 1; den[2] = 2; num[n_?EvenQ] := num[n] = num[n/2]; den[n_?EvenQ] := den[n] = num[n/2] + den[n/2]; num[n_?OddQ] := num[n] = den[(n-1)/2]; den[n_?OddQ] := den[n] = num[(n-1)/2] + den[(n-1)/2]; A093875 = Table[den[n], {n, 1, 83}] (* Jean-Fran├žois Alcover, Dec 16 2011 *)

CROSSREFS

The numerators are in A093873. Usually one only considers the left-hand half of the tree, which gives the fractions A020651/A086592. See A086592 for more information, references to Kepler, etc.

Cf. A071585, A093873

Sequence in context: A101402 A156251 A116458 * A266193 A114214 A321318

Adjacent sequences:  A093872 A093873 A093874 * A093876 A093877 A093878

KEYWORD

nonn,easy,frac

AUTHOR

N. J. A. Sloane and Reinhard Zumkeller, May 24 2004

STATUS

approved

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Last modified January 16 15:53 EST 2019. Contains 319195 sequences. (Running on oeis4.)