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A093873
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Numerators in Kepler's tree of harmonic fractions.
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13
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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, 5, 4, 5, 3, 7, 4, 7, 2, 7, 5, 7, 3, 8, 5, 8, 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, 6
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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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).
a(A029744(n-1)) = 1; a(A070875(n-1)) = 2; a(A123760(n-1)) = 3. - Reinhard Zumkeller, Oct 13 2006
a011782(k) = SUM(a(n)/A093875(n): 2^k<=n<2^(k+1)), k>=0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 17 2010]
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LINKS
| R. Zumkeller, Table of n, a(n) for n = 1..10000
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FORMULA
| a(n) = a([n/2])*(1 - n mod 2) + A093875([n/2])*(n mod 2).
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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
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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]; A093873 = Table[num[n], {n, 1, 97}] (* From Jean-François Alcover, Dec 16 2011 *)
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PROG
| Contribution from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 17 2010: (Start)
(Other) Haskell:
{-# LANGUAGE ViewPatterns #-}
import Data.Ratio((%), numerator, denominator)
rat :: Rational -> (Integer, Integer)
rat r = (numerator r, denominator r)
data Harmony = Harmony Harmony Rational Harmony
rows :: Harmony -> [[Rational]]
rows (Harmony hL r hR) = [r] : zipWith (++) (rows hL) (rows hR)
kepler :: Rational -> Harmony
kepler r = Harmony (kepler (i%(i+j))) r (kepler (j%(i+j)))
.......... where (rat -> (i, j)) = r
-- Full tree of Kepler's harmonic fractions:
k = rows $ kepler 1 :: [[Rational]] -- as list of lists
h = concat k :: [Rational] -- flattened
a093873 n = numerator $ h !! (n - 1)
a093875 n = denominator $ h !! (n - 1)
a011782 n = numerator $ (map sum k) !! n -- denominator == 1
-- length (k !! n) == 2^n
-- numerator $ (map last k) !! n == fibonacci (n + 1)
-- denominator $ (map last k) !! n == fibonacci (n + 2)
-- numerator $ (map maximum k) !! n == n
-- denominator $ (map maximum k) !! n == n + 1
-- eop. (End)
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CROSSREFS
| The denominators are in A093875. 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.
Sequence in context: A112309 A160006 A060682 * A161148 A143773 A191372
Adjacent sequences: A093870 A093871 A093872 * A093874 A093875 A093876
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KEYWORD
| nonn,easy,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com) and Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 24 2004
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