This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A007305 M0113 #111 Aug 03 2024 01:49:05
%S A007305 0,1,1,1,2,1,2,3,3,1,2,3,3,4,5,5,4,1,2,3,3,4,5,5,4,5,7,8,7,7,8,7,5,1,
%T A007305 2,3,3,4,5,5,4,5,7,8,7,7,8,7,5,6,9,11,10,11,13,12,9,9,12,13,11,10,11,
%U A007305 9,6,1,2,3,3,4,5,5,4,5,7,8,7,7,8,7,5,6,9,11,10,11,13,12,9,9,12,13,11
%N A007305 Numerators of Farey (or Stern-Brocot) tree fractions.
%C A007305 From _Yosu Yurramendi_, Jun 25 2014: (Start)
%C A007305 If the terms (n>0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...
%C A007305 1,
%C A007305 1,2,
%C A007305 1,2,3,3,
%C A007305 1,2,3,3,4,5,5,4,
%C A007305 1,2,3,3,4,5,5,4,5,7,8,7,7,8,7,5,
%C A007305 1,2,3,3,4,5,5,4,5,7,8,7,7,8,7,5,6,9,11,10,11,13,12,9,9,12,13,11,10,11,9,6,
%C A007305 then the sum of the m-th row is 3^m (m = 0,1,2,), each column k is constant, and the constants are from A007306, denominators of Farey (or Stern-Brocot) tree fractions (see formula).
%C A007305 If the rows are written in a right-aligned fashion:
%C A007305 1,
%C A007305 1,2,
%C A007305 1, 2,3,3,
%C A007305 1, 2, 3, 3, 4, 5,5,4,
%C A007305 1,2, 3, 3, 4, 5, 5,4,5, 7, 8, 7, 7, 8,7,5,
%C A007305 1,2,3,3,4,5,5,4,5,7,8,7,7,8,7,5,6,9,11,10,11,13,12,9,9,12,13,11,10,11,9,6,
%C A007305 then each column is an arithmetic sequence. The differences of the arithmetic sequences also give the sequence A007306 (see formula). The first terms of columns are from A007305 itself (a(A004761(n+1)) = a(n), n>0), and the second ones from A049448 (a(A004761(n+1)+2^A070941(n)) = A049448(n), n>0). (End)
%C A007305 If the sequence is considered in blocks of length 2^m, m = 0,1,2,..., the blocks are the reverse of the blocks of A047679: (a(2^m+1+k) = A047679(2^(m+1)-2-k), m = 0,1,2,..., k = 0,1,2,...,2^m-1). - _Yosu Yurramendi_, Jun 30 2014
%D A007305 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 117.
%D A007305 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 23.
%D A007305 J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
%D A007305 W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
%D A007305 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 141.
%D A007305 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007305 T. D. Noe, <a href="/A007305/b007305.txt">Table of n, a(n) for n=0..4096</a>
%H A007305 A. Bogomolny, <a href="http://www.cut-the-knot.org/blue/Stern.shtml">Stern-Brocot Tree</a>
%H A007305 A. Bogomolny, <a href="http://www.cut-the-knot.org/blue/SB_props.shtml">Inspiration for Maple code</a>
%H A007305 A. Brocot, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k1661912">Calcul des rouages par approximation, nouvelle méthode</a>, Revue Chonométrique 3, 186-194, 1861.
%H A007305 G. A. Jones, <a href="http://www.mat.univie.ac.at/~slc/opapers/s18jones.html">The Farey graph</a>, Séminaire Lotharingien de Combinatoire, B18e (1987), 2 pp.
%H A007305 Shin-ichi Katayama, <a href="https://www-math.ias.tokushima-u.ac.jp/journal/2013/Katayama47.pdf">Modified Farey trees and Pythagorean triples</a>, Journal of mathematics, the University of Tokushima, 47, 2013.
%H A007305 G. Melançon, <a href="https://doi.org/10.1016/S0012-365X(99)00123-5">Lyndon factorization of sturmian words</a>, Discr. Math., 210 (2000), 137-149.
%H A007305 Hugo Pfoertner, <a href="https://oeis.org/plot2a?name1=A007305&name2=A007306&tform1=untransformed&tform2=untransformed&shift=0&radiop1=ratio&drawpoints=true">Ratio A007305(n)/A007306(n) vs n</a>, using Plot 2.
%H A007305 N. J. A. Sloane, <a href="/stern_brocot.html">Stern-Brocot or Farey Tree</a>
%H A007305 Noam Zimhoni, <a href="https://arxiv.org/abs/1904.11782">A forest of Eisensteinian triplets</a>, arXiv:1904.11782 [math.NT], 2019.
%H A007305 <a href="/index/Fo#fraction_trees">Index entries for fraction trees</a>
%H A007305 <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%F A007305 a(n) = SternBrocotTreeNum(n-1) # n starting from 2 gives the sequence from 1, 1, 2, 1, 2, 3, 3, 1, 2, 3, 3, 4, 5, 5, 4, 1, ...
%F A007305 From _Reinhard Zumkeller_, Dec 22 2008: (Start)
%F A007305 For n > 1: a(n+2) = if A025480(n-1) != 0 and A025480(n) != 0 then a(A025480(n-1)+2) + a(A025480(n)+2) else if A025480(n)=0 then a(A025480(n-1)+2)+1 else 0 + a(A025480(n-1)+2).
%F A007305 a(A054429(n)+2) = A047679(n).
%F A007305 a(n+2) = A047679(A054429(n)).
%F A007305 A153036(n+1) = floor(a(n+2)/A047679(n)). (End)
%F A007305 From _Yosu Yurramendi_, Jun 25 2014: (Start)
%F A007305 For m = 1,2,3,..., and k = 0,1,2,...,2^(m-1)-1, with a(1)=1:
%F A007305 a(2^m+k) = a(2^(m-1)+k);
%F A007305 a(2^m+2^(m-1)+k) = a(2^(m-1)+k) + a(2^m-k-1). (End)
%F A007305 a(2^(m+2)-k) = A007306(2^(m+1)-k), m=0,1,2,..., k=0,1,2,...,2^m-1. - _Yosu Yurramendi_, Jul 04 2014
%F A007305 a(2^(m+1)+2^m+k) - a(2^m+k) = A007306(2^m-k+1), m=1,2,..., k=1,2,...,2^(m-1). - _Yosu Yurramendi_, Jul 05 2014
%F A007305 From _Yosu Yurramendi_, Jan 01 2015: (Start)
%F A007305 a(2^m+2^q-1) = q+1, q = 0, 1, 2,..., m = q, q+1, q+2,...
%F A007305 a(2^m+2^q) = q+2, q = 0, 1, 2,..., m = q+1, q+2, q+3,... (End)
%F A007305 a(2^m+k) = A007306(k+1), m >= 0, 0 <= k < 2*m. - _Yosu Yurramendi_, May 20 2019
%F A007305 a(n) = A002487(A059893(n)), n > 0. - _Yosu Yurramendi_, Sep 29 2021
%e A007305 A007305/A007306 = [ 0/1; 1/1; ] 1/2; 1/3, 2/3; 1/4, 2/5, 3/5, 3/4; 1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, ...
%e A007305 Another version of Stern-Brocot is A007305/A047679 = 1, 2, 1/2, 3, 1/3, 3/2, 2/3, 4, 1/4, 4/3, 3/4, 5/2, 2/5, 5/3, 3/5, 5, 1/5, 5/4, 4/5, ...
%p A007305 A007305 := proc(n) local b; b := proc(n) option remember; local msb, r;
%p A007305 if n < 3 then return 1 fi; msb := ilog2(n); r := n - 2^msb;
%p A007305 if ilog2(r) = msb - 1 then b(r) + b(3*2^(msb-1) - r - 1) else b(2^(msb - 1) + r) fi end: if n = 0 then 0 else b(n-1) fi end: # _Antti Karttunen_, Mar 19 2000 [Corrected and rewritten by _Peter Luschny_, Apr 24 2024]
%p A007305 seq(A007305(n), n = 0..92);
%t A007305 sbt[n_] := Module[{R,L,Y}, R={{1,0},{1,1}}; L={{1,1},{0,1}}; Y={{1,0},{0,1}}; w[b_] := Fold[ #1.If[ #2 == 0,L,R] &,Y,b]; u[a_] := {a[[2,1]]+a[[2,2]],a[[1,1]]+a[[1,2]]}; Map[u,Map[w,Tuples[{0,1},n]]]]
%t A007305 A007305(n) = Flatten[Append[{0,1},Table[Map[First,sbt[i]],{i,0,5}]]]
%t A007305 A047679(n) = Flatten[Table[Map[Last,sbt[i]],{i,0,5}]]
%t A007305 (* _Peter Luschny_, Apr 27 2009 *)
%o A007305 (R)
%o A007305 a <- 1
%o A007305 for(m in 1:6) for(k in 0:(2^(m-1)-1)) {
%o A007305 a[2^m+ k] <- a[2^(m-1)+k]
%o A007305 a[2^m+2^(m-1)+k] <- a[2^(m-1)+k] + a[2^m-k-1]
%o A007305 }
%o A007305 a
%o A007305 # _Yosu Yurramendi_, Jun 25 2014
%Y A007305 Cf. A007306, A006842, A006843, A047679, A054424, A057114, A152975.
%K A007305 nonn,frac,tabf,nice,look
%O A007305 0,5
%A A007305 _N. J. A. Sloane_