A007305 Numerators of Farey (or Stern-Brocot) tree fractions. (Formerly M0113)

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, 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, 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

Offset

0,5

Comments

From Yosu Yurramendi, Jun 25 2014: (Start)

If the terms (n>0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...

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,

1,2,3,3,4,5,5,4,5,7,8,7,7,8,7,5,6,9,11,10,11,13,12,9,9,12,13,11,10,11,9,6,

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).

If the rows are written in a right-aligned fashion:

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,

1,2,3,3,4,5,5,4,5,7,8,7,7,8,7,5,6,9,11,10,11,13,12,9,9,12,13,11,10,11,9,6,

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)

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

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 117.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 23.

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.

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.

I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 141.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n=0..4096

A. Bogomolny, Stern-Brocot Tree

A. Bogomolny, Inspiration for Maple code

A. Brocot, Calcul des rouages par approximation, nouvelle méthode, Revue Chonométrique 3, 186-194, 1861.

G. A. Jones, The Farey graph, Séminaire Lotharingien de Combinatoire, B18e (1987), 2 pp.

Shin-ichi Katayama, Modified Farey trees and Pythagorean triples, Journal of mathematics, the University of Tokushima, 47, 2013.

G. Melançon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137-149.

Hugo Pfoertner, Ratio A007305(n)/A007306(n) vs n, using Plot 2.

N. J. A. Sloane, Stern-Brocot or Farey Tree

Noam Zimhoni, A forest of Eisensteinian triplets, arXiv:1904.11782 [math.NT], 2019.

Index entries for fraction trees

Index entries for sequences related to Stern's sequences

Formula

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, ...

From Reinhard Zumkeller, Dec 22 2008: (Start)

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).

a(A054429(n)+2) = A047679(n).

a(n+2) = A047679(A054429(n)).

A153036(n+1) = floor(a(n+2)/A047679(n)). (End)

From Yosu Yurramendi, Jun 25 2014: (Start)

For m = 1,2,3,..., and k = 0,1,2,...,2^(m-1)-1, with a(1)=1:

a(2^m+k) = a(2^(m-1)+k);

a(2^m+2^(m-1)+k) = a(2^(m-1)+k) + a(2^m-k-1). (End)

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

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

From Yosu Yurramendi, Jan 01 2015: (Start)

a(2^m+2^q-1) = q+1, q = 0, 1, 2,..., m = q, q+1, q+2,...

a(2^m+2^q) = q+2, q = 0, 1, 2,..., m = q+1, q+2, q+3,... (End)

a(2^m+k) = A007306(k+1), m >= 0, 0 <= k < 2*m. - Yosu Yurramendi, May 20 2019

a(n) = A002487(A059893(n)), n > 0. - Yosu Yurramendi, Sep 29 2021

Example

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, ...
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, ...

Maple

A007305 := proc(n) local b; b := proc(n) option remember; local msb, r;
if n < 3 then return 1 fi; msb := ilog2(n); r := n - 2^msb;
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]
seq(A007305(n), n = 0..92);

Mathematica

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]]]]
A007305(n) = Flatten[Append[{0, 1}, Table[Map[First, sbt[i]], {i, 0, 5}]]]
A047679(n) = Flatten[Table[Map[Last, sbt[i]], {i, 0, 5}]]
(* Peter Luschny, Apr 27 2009 *)

Prog

(R)
a <- 1
for(m in 1:6) for(k in 0:(2^(m-1)-1)) {
  a[2^m+        k] <- a[2^(m-1)+k]
  a[2^m+2^(m-1)+k] <- a[2^(m-1)+k] + a[2^m-k-1]
}
a
# Yosu Yurramendi, Jun 25 2014

Crossrefs

Cf. A007306, A006842, A006843, A047679, A054424, A057114, A152975.

Sequence in context: A035531 A118977 A071766 * A112531 A100002 A348330

Adjacent sequences: A007302 A007303 A007304 * A007306 A007307 A007308

Keyword

nonn,frac,tabf,nice,look,changed

Author

N. J. A. Sloane

Status

approved