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%I #59 Apr 21 2024 22:13:57
%S 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,
%T 13,7,7,11,11,13,13,14,14,13,13,17,17,15,15,18,18,11,11,16,16,17,17,
%U 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
%N Denominators in left-hand half of Kepler's tree of fractions.
%C 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).
%C Level n of the left-hand half of the tree consists of 2^(n-1) 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; ... .
%C The right-hand half is identical to the left-hand half. - _Michel Dekking_, Oct 05 2017
%C n>1 occurs in this sequence phi(n) = A000010(n) times, as it occurs in A007306 (_Franklin T. Adams-Watters_' comment), that is the sequence obtained by adding numerator and denominator in the Calkin-Wilf enumeration system of positive rationals. A020650(n)/A020651(n) is also an enumeration system of all positive rationals (Yu-Ting 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
%D Johannes Kepler, Mysterium cosmographicum, Tuebingen, 1596, 1621, Caput XII.
%D Johannes Kepler, Harmonice Mundi, Linz, 1619, Liber III, Caput II.
%D 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.
%H Johannes Kepler, <a href="http://archive.org/details/ioanniskepplerih00kepl">Harmonices mundi libri V ...</a> (A Latin original scanned in Internet Archive. The fraction-tree is illustrated on the page 27 of the third book (Liber III), which is on the page 117 of the PDF-document.)
%H 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> (Illustrates the A020651/A086592-tree).
%H OEIS Wiki, <a href="http://oeis.org/wiki/Historical_sequences">Historical sequences</a>
%H Pelegrí Viader, Jaume Paradís and Lluís Bibiloni, <a href="https://doi.org/10.1006/jnth.1998.2294">A New Light on Minkowski's ?(x) Function</a>, J. Number Theory, 73 (2) (1998), 212-227. See p. 215.
%H <a href="/index/Fo#fraction_trees">Index entries for fraction trees</a>
%F a(n) = A020650(n) + A020651(n) = A020650(2n).
%F a(n) = A071585(A059893(n)), a(A059893(n)) = A071585(n), n > 0. - _Yosu Yurramendi_, May 30 2017
%F a(2*n-1) = A086593(n); a(2*n) = A086593(n+1), n > 0. - _Yosu Yurramendi_, May 16 2018
%F a(n) = A007306(A231551(n)), n > 0. - _Yosu Yurramendi_, Aug 07 2021
%t (* 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] (* _Jean-François Alcover_, Jan 22 2016 *)
%o (R)
%o maxlevel <- 15
%o d <- c(1,2)
%o for(m in 0:maxlevel)
%o for(k in 1:2^m) {
%o d[2^(m+1) +k] <- d[k] + d[2^m+k]
%o d[2^(m+1)+2^m+k] <- d[2^(m+1)+k]
%o }
%o b <- vector()
%o for(m in 0:maxlevel) for(k in 0:(2^m-1)) b[2^m+k] <- d[2^(m+1)+k]
%o a <- vector()
%o for(n in 1:2^maxlevel) {a[2*n-1] <- b[n]; a[2*n] <- b[n+1]}
%o a[1:128]
%o # _Yosu Yurramendi_, May 16 2018
%o (PARI) b(n) = my(b=binary(n)); fromdigits(concat(b[1], Vecrev(vector(#b-1, k, b[k+1]))), 2); \\ from A059893
%o a(n) = my(n=b(n), x=1, y=1); for(i=0, logint(n, 2), if(bittest(n, i), [x, y]=[x+y, x], [x, y]=[x, x+y])); x \\ _Mikhail Kurkov_, Mar 11 2023 [verification needed]
%Y Bisection of A020650.
%Y See A093873/A093875 for the full tree.
%Y A020651 gives the numerators. Bisection: A086593. Cf. A002487, A004169.
%K nonn,frac,tabf,changed
%O 1,1
%A _Antti Karttunen_, Aug 28 2003
%E Entry revised by _N. J. A. Sloane_, May 24 2004
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