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

%I A294442 #34 Dec 26 2024 10:15:37
%S A294442 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,
%T A294442 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,
%U A294442 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
%N A294442 Kepler's tree of fractions, read across rows (the fraction i/j is represented as the pair i,j).
%C A294442 The first row contains the single fraction 1/1,
%C A294442 the second row contains the single fraction 1/2,
%C A294442 and thereafter below each fraction i/j we write two fractions i/(i+j), j/(i+j).
%C A294442 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.
%C A294442 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.
%H A294442 Michael De Vlieger, <a href="/A294442/b294442.txt">Table of n, a(n) for n = 0..16383</a>
%H A294442 Johannes Kepler, <a href="http://web.archive.org/web/20081009062459/http://ndirty.cute.fi/~karttu/Kepler/a086592.htm">Excerpt from the Chapter II of the Book III of the Harmony of the World: On the seven harmonic divisions of the string</a>.
%H A294442 Richard J. Mathar, <a href="/A294443/a294443.pdf">The Kepler binary tree of reduced fractions</a>, 2017.
%H A294442 <a href="/index/Fo#fraction_trees">Index entries for fraction trees</a>.
%H A294442 <a href="/index/Mu#music">Index entries for sequences related to music</a>.
%e A294442 The tree begins as follows:
%e A294442 ..............1/1
%e A294442 ...............|
%e A294442 ..............1/2
%e A294442 .........../.......\
%e A294442 ......1/3.............2/3
%e A294442 ...../....\........../...\
%e A294442 ..1/4.....3/4.....2/5.....3/5
%e A294442 ../..\..../..\..../..\..../..\
%e A294442 1/5.4/5.3/7.4/7.2/7.5/7.3/8.5/8
%p A294442 # S[n] is the list of fractions, written as pairs [i,j], in row n of Kepler's triangle
%p A294442 S[0]:=[[1,1]]; S[1]:=[[1,2]];
%p A294442 for n from 2 to 10 do
%p A294442 S[n]:=[];
%p A294442 for k from 1 to nops(S[n-1]) do
%p A294442 t1:=S[n-1][k];
%p A294442 a:=[t1[1],t1[1]+t1[2]];
%p A294442 b:=[t1[2],t1[1]+t1[2]];
%p A294442 S[n]:=[op(S[n]),a,b];
%p A294442 od:
%p A294442 lprint(S[n]);
%p A294442 od:
%t A294442 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 *)
%Y A294442 Cf. A020651, A086593.
%Y A294442 A different version of the Kepler tree is described in A093873.
%Y A294442 See A294446 for the tree of Farey fractions.
%K A294442 nonn,tabf
%O A294442 0,4
%A A294442 _N. J. A. Sloane_, Nov 20 2017