|
| |
|
|
A093875
|
|
Denominators in Kepler's tree of harmonic fractions.
|
|
7
| |
|
|
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, 6, 6, 9, 9, 10, 10, 11, 11, 9, 9, 12, 12, 11, 11, 13, 13, 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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
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
|
|
|
FORMULA
| a(n) = a([n/2]) + A093873([n/2]).
|
|
|
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
|
|
|
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]; A093875 = Table[den[n], {n, 1, 83}] (* From Jean-François Alcover, Dec 16 2011 *)
|
|
|
CROSSREFS
| The numerators 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: A101402 A156251 A116458 * A114214 A196383 A074198
Adjacent sequences: A093872 A093873 A093874 * A093876 A093877 A093878
|
|
|
KEYWORD
| nonn,easy,frac
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com) and Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 24 2004
|
| |
|
|
