This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A093873 Numerators in Kepler's tree of harmonic fractions. 16
 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; text; internal format)
 OFFSET 1,5 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). LINKS R. Zumkeller, Table of n, a(n) for n = 1..10000 Johannes Kepler, Harmonices Mundi, Liber III, see p. 27. FORMULA a(n) = a([n/2])*(1 - n mod 2) + A093875([n/2])*(n mod 2). 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. [Reinhard Zumkeller, Oct 17 2010] a(1) =  1. For all n>0  a(2n) =  a(n), a(2n+1) =  A093875(n). - Yosu Yurramendi, Jan 09 2016 a(4n+3) = a(4n+1), a(4n+2) = a(4n+1) - a(4n), a(4n+1) = A071585(n). - Yosu Yurramendi, Jan 11 2016 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+1)+2^m+k) = A020651(2^m+k), m>=0, 0<=k<2^m. - Yosu Yurramendi, May 18 2016 a(k) = A020651(2^(m+1)+k) - A020651(2^m+k), m>0, 0=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-1-(2^r -1)) = A000045(m-r), m >= 1, 0 <= r <= m-1. - 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 A093875(2n) - a(2n) = A093875(n), n > 0; A093875(2n+1) - a(2n+1) = a(n), n > 0. - Yosu Yurramendi, Jul 23 2016 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 MAPLE 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 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}] (* Jean-François Alcover, Dec 16 2011 *) PROG (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. -- Reinhard Zumkeller, Oct 17 2010 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. See A294442 for another version of Kepler's tree of fractions. Sequence in context: A280363 A217743 A238845 * A305974 A161148 A143773 Adjacent sequences:  A093870 A093871 A093872 * A093874 A093875 A093876 KEYWORD nonn,easy,frac,look,hear AUTHOR N. J. A. Sloane and Reinhard Zumkeller, May 24 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 17 15:01 EDT 2019. Contains 328116 sequences. (Running on oeis4.)