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Search: id:A086592
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| A086592 |
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Denominators in left-hand half of Kepler's tree of fractions. |
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+0
6
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| 2, 3, 3, 4, 4, 5, 5, 5, 5, 7, 7, 7, 7, 8, 8, 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, 17, 17, 19, 19, 14, 14, 19, 19, 18, 18, 21, 21, 8, 8, 13, 13, 16, 16, 17, 17, 17, 17, 22, 22, 19, 19, 23
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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).
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; ...
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REFERENCES
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Johannes Kepler, Mysterium cosmographicum, Tuebingen, 1596, 1621, Caput XII.
Johannes Kepler, Harmonice Mundi, Linz, 1619, Liber III, Caput II.
Johannes Kepler, The Harmony of the World [1619], trans. E. J. Aiton, A. M. Duncan and J. V. Field, American Philosophical Society, Philadelphia, 1997, p. 163.
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LINKS
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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).
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CROSSREFS
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Bisection of A020650.
See A093873/A093875 for the full tree.
a(n) = A020650(n)+A020651(n) = A020650(2n). A020651 gives the numerators. Bisection: A086593. Cf. A002487, A004169.
Adjacent sequences: A086589 A086590 A086591 this_sequence A086593 A086594 A086595
Sequence in context: A038567 A036234 A061091 this_sequence A132663 A023964 A000267
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KEYWORD
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nonn,frac,tabf
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AUTHOR
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Antti Karttunen (his_firstname.his_surname(AT)iki.fi) Aug 28 2003
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EXTENSIONS
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Entry revised by njas, May 24 2004
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