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A093873
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Numerators in Kepler's tree of harmonic fractions.
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15
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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
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OFFSET
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1,5
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COMMENTS
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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).
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LINKS
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FORMULA
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a(n) = a([n/2])*(1 - n mod 2) + A093875([n/2])*(n mod 2).
G.f. G(x) satisfies G(x) = x + (1+x) G(x^2) + Sum_{k>=2} x (1+x^(2^(k-1))) G(x^(2^k)). - Robert Israel, Jan 11 2016
a(2^(m+1)+k) - a(2^m+k) = a(k) , m >=0, 0 <= k < 2^m. For k=0 a(0)=0 is needed. - Yosu Yurramendi, Jul 22 2016
a(2^(m+2)-1-k) = a(2^(m+1)-1-k) + a(2^m-1-k), m >= 1, 0 <= k < 2^m. - Yosu Yurramendi, Jul 22 2016
a(2^m+2^r) = m-r, , m >= 1, 0 <= r <= m-1 ; a(2^m+2^r+2^(r-1)) = m-(r-1), m >= 2, 0 <= r <= m-1. - Yosu Yurramendi, Jul 22 2016
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EXAMPLE
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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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MAPLE
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M:= 8: # to get a(1) .. a(2^M-1)
gen[1]:= [1];
for n from 2 to M do
gen[n]:= map(t -> (numer(t)/(numer(t)+denom(t)),
denom(t)/(numer(t)+denom(t))), gen[n-1]);
od:
seq(op(map(numer, gen[i])), i=1..M): # Robert Israel, Jan 11 2016
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MATHEMATICA
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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}] (* Jean-François Alcover, Dec 16 2011 *)
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PROG
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(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.
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CROSSREFS
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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.
See A294442 for another version of Kepler's tree of fractions.
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KEYWORD
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AUTHOR
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STATUS
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approved
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