

A086592


Denominators in lefthand half of Kepler's tree of fractions.


12



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
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OFFSET

1,1


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).
Level n of the lefthand half of the tree consists of 2^(n1) 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; ... .
The righthand half is identical to the lefthand half.  Michel Dekking, Oct 05 2017
n>1 occurs in this sequence phi(n) = A000010(n) times, as it occurs in A007306 (Franklin T. AdamsWatters' comment), that is the sequence obtained by adding numerator and denominator in the CalkinWilf enumeration system of positive rationals. A020650(n)/A020651(n) is also an enumeration system of all positive rationals (YuTing system), and in each level m >= 0 (ranks between 2^m and 2^(m+1)1) rationals are the same in both systems. Thus a(n) has the same terms in each level as A007306. The same property occurs in all numerator+denominator sequences of enumeration systems of positive rationals, as, for example, A007306 (A007305+A047679), A071585 (A229742+A071766), and A268087 (A162909+A162910).  Yosu Yurramendi, Apr 06 2016


REFERENCES

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.


LINKS

Table of n, a(n) for n=1..78.
Johannes Kepler, Harmonices mundi libri V ... (A Latin original scanned in Internet Archive. The fractiontree is illustrated on the page 27 of the third book (Liber III), which is on the page 117 of the PDFdocument.)
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/A086592tree).
OEIS Wiki, Historical sequences
Index entries for fraction trees


FORMULA

a(n) = A020650(n) + A020651(n) = A020650(2n).
a(n) = A071585(A059893(n)), a(A059893(n)) = A071585(n), n > 0.  Yosu Yurramendi, May 30 2017
a(2*n1) = A086593(n); a(2*n) = A086593(n+1), n > 0.  Yosu Yurramendi, May 16 2018


MATHEMATICA

(* b = A020650 *) b[1] = 1; b[2] = 2; b[3] = 1; b[n_] := b[n] = Switch[ Mod[n, 4], 0, b[n/2 + 1] + b[n/2], 1, b[(n  1)/2 + 1], 2, b[(n  2)/2 + 1] + b[(n  2)/2], 3, b[(n  3)/2]]; a[n_] := b[2n]; Array[a, 100] (* JeanFrançois Alcover, Jan 22 2016 *)


PROG

(R)
maxlevel < 15
d < c(1, 2)
for(m in 0:maxlevel)
for(k in 1:2^m) {
d[2^(m+1) +k] < d[k] + d[2^m+k]
d[2^(m+1)+2^m+k] < d[2^(m+1)+k]
}
b < vector()
for(m in 0:maxlevel) for(k in 0:(2^m1)) b[2^m+k] < d[2^(m+1)+k]
a < vector()
for(n in 1:2^maxlevel) {a[2*n1] < b[n]; a[2*n] < b[n+1]}
a[1:128]
# Yosu Yurramendi, May 16 2018


CROSSREFS

Bisection of A020650.
See A093873/A093875 for the full tree.
A020651 gives the numerators. Bisection: A086593. Cf. A002487, A004169.
Sequence in context: A036234 A061091 A196241 * A279783 A132663 A306631
Adjacent sequences: A086589 A086590 A086591 * A086593 A086594 A086595


KEYWORD

nonn,frac,tabf


AUTHOR

Antti Karttunen, Aug 28 2003


EXTENSIONS

Entry revised by N. J. A. Sloane, May 24 2004


STATUS

approved



