Search: id:A020651
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%I A020651
%S A020651 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,
%T A020651 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,
%U A020651 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
%N 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)).
%C A020651 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.
%C A020651 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 (Antti.Karttunen(AT)iki.fi), May 26 2004
%H A020651 T. D. Noe, <a href="b020651.txt">Table of n, a(n) for n=1..10000</a>
%H A020651 Johannes Kepler, <a href="http://www.iki.fi/~kartturi/Kepler/a086592.htm">
               Excerpt from the Chapter II of the Book III of the Harmony of the 
               World: On the seven harmonic divisions of the string</a> (illustrates 
               the A020651/A086592-tree).
%F A020651 a(1) = 1, a(2n) = a(n), a(2n+1) = A020650(2n) - Antti Karttunen (Antti.Karttunen(AT)iki.fi), 
               May 26 2004
%e A020651 1, 2, 1/2, 3, 1/3, 3/2, 2/3, 4, 1/4, 4/3, ...
%p A020651 A020651 := n -> `if`((n < 2),n,`if`(type(n,even), A020651(n/2), A020650(n-1)));
%Y A020651 See A093873/A093875 for the full Kepler tree.
%Y A020651 Cf. A020650, A086592.
%Y A020651 Sequence in context: A038568 A071912 A070940 this_sequence A160232 A002487 
               A060162
%Y A020651 Adjacent sequences: A020648 A020649 A020650 this_sequence A020652 A020653 
               A020654
%K A020651 nonn,easy,frac,nice
%O A020651 1,3
%A A020651 David W. Wilson (davidwwilson(AT)comcast.net)
%E A020651 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), May 24 2004

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