A294442 Kepler's tree of fractions, read across rows (the fraction i/j is represented as the pair i,j).

1, 1, 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, 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, 7, 6, 7, 5, 11, 6, 11, 4, 13, 9, 13, 5, 14, 9, 14, 3, 13, 10, 13, 7, 17, 10, 17, 4, 15, 11, 15, 7, 18, 11

Offset

0,4

Comments

The first row contains the single fraction 1/1,

the second row contains the single fraction 1/2,

and thereafter below each fraction i/j we write two fractions i/(i+j), j/(i+j).

If we just look at the numerators we recover the same sequence, and if we just look at the denominators we get A086593 with the terms (after the first) repeated.

Sequence A020651 is almost the same as this, except that it lacks one of the initial 1's, and the definition focuses on single numbers rather than pairs of numbers or fractions. For that reason it seems to be best to have a separate entry (this sequence) for the actual tree.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..16383

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.

Richard J. Mathar, The Kepler binary tree of reduced fractions, 2017.

Example

The tree begins as follows:
..............1/1
...............|
..............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

Maple

# S[n] is the list of fractions, written as pairs [i, j], in row n of Kepler's triangle
S[0]:=[[1, 1]]; S[1]:=[[1, 2]];
for n from 2 to 10 do
S[n]:=[];
for k from 1 to nops(S[n-1]) do
t1:=S[n-1][k];
a:=[t1[1], t1[1]+t1[2]];
b:=[t1[2], t1[1]+t1[2]];
S[n]:=[op(S[n]), a, b];
od:
lprint(S[n]);
od:

Mathematica

Map[{Numerator@ #, Denominator@ #} &, #] &@ Flatten@ Nest[Append[#, Flatten@ Map[{#1/(#1 + #2), #2/(#1 + #2)} & @@ {Numerator@ #, Denominator@ #} &, Last@ #]] &, {{1/1}, {1/2}}, 5] // Flatten (* Michael De Vlieger, Apr 18 2018 *)

Crossrefs

Cf. A020651, A086593.

A different version of the Kepler tree is described in A093873.

See A294446 for the tree of Farey fractions.

Sequence in context: A071912 A070940 A020651 * A281392 A287051 A368147

Adjacent sequences: A294439 A294440 A294441 * A294443 A294444 A294445

Keyword

nonn,tabf

Author

N. J. A. Sloane, Nov 20 2017

Status

approved