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A093875 Denominators in Kepler's tree of harmonic fractions. 8

%I

%S 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,

%T 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,

%U 11,13,13,7,7,11,11,13,13,14,14,13,13,17,17,15,15,18,18,11,11,16,16

%N Denominators in Kepler's tree of harmonic fractions.

%C 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).

%C It appears that A071585 is a bisection of this sequence, which itself is a bisection of A093873. - _Yosu Yurramendi_, Jan 09 2016

%H R. Zumkeller, <a href="/A093875/b093875.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Fo#fraction_trees">Index entries for fraction trees</a>

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

%F 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

%F a(2n) - A093873(2n) = a(n), n > 0; a(2n+1) - A093873(2n+1) = A093873(n), n > 0. - _Yosu Yurramendi_, Jul 23 2016

%F From _Yosu Yurramendi_, Jul 25 2016: (Start)

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

%F 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.

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

%e The first few fractions are:

%e 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

%e - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ...

%e 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

%t 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 *)

%Y 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.

%Y Cf. A071585, A093873

%K nonn,easy,frac

%O 1,2

%A _N. J. A. Sloane_ and _Reinhard Zumkeller_, May 24 2004

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Last modified March 23 16:52 EDT 2019. Contains 321432 sequences. (Running on oeis4.)