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

Author

N. J. A. Sloane, Nov 20 2017

Keywords

• nonn
• tabf

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.

Examples

			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
		

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.
• Index entries for fraction trees.
• Index entries for sequences related to music.

Crossrefs

Cf. A020651, A086593.

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

See A294446 for the tree of Farey fractions.

Programs

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 *)