A152976 Denominators of the redundant Stern-Brocot structure; numerators=A152975.
1, 2, 3, 1, 3, 6, 3, 5, 2, 3, 1, 4, 9, 5, 10, 5, 9, 4, 7, 3, 6, 3, 5, 2, 3, 1, 5, 12, 7, 15, 8, 15, 7, 14, 7, 15, 8, 15, 7, 12, 5, 9, 4, 9, 5, 10, 5, 9, 4, 7, 3, 6, 3, 5, 2, 3, 1, 6, 15, 9, 20, 11, 21, 10, 21, 11, 24, 13, 25, 12, 21, 9, 18, 9, 21, 12, 25, 13, 24, 11, 21, 10, 21, 11, 20, 9, 15, 6
Offset: 1
A007305 Numerators of Farey (or Stern-Brocot) tree fractions.
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
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
Examples
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, ...
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
Programs
-
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);
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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 *) -
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
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).
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
A060188 A column and diagonal of A060187.
1, 6, 23, 76, 237, 722, 2179, 6552, 19673, 59038, 177135, 531428, 1594309, 4782954, 14348891, 43046704, 129140145, 387420470, 1162261447, 3486784380, 10460353181, 31381059586, 94143178803, 282429536456, 847288609417
Offset: 2
Comments
Sums of rows of the numerators and of the denominators of the redundant Stern-Brocot structure A152975/A152976: a(n+2) = Sum_{k=2^n..(2^(n+1) -1)} A152975(k) = Sum_{k=2^n..(2^(n+1) -1)} A152976(k). - Reinhard Zumkeller, Dec 22 2008
Links
- Vincenzo Librandi, Table of n, a(n) for n = 2..2000
- P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1920), 305-340; Coll. Papers II, pp. 267-302.
- Index entries for linear recurrences with constant coefficients, signature (5,-7,3).
Programs
-
Magma
[3^(n-1)-n: n in [2..30]]; // Vincenzo Librandi, Sep 05 2011
-
Maple
a[0]:=1:for n from 1 to 24 do a[n]:=(4*a[n-1]-3*a[n-2]+2) od: seq(a[n], n=0..24); # Zerinvary Lajos, Jun 08 2007
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Mathematica
Table[3^(n-1) -n, {n,2,30}] (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *) LinearRecurrence[{5,-7,3},{1,6,23},30] (* Harvey P. Dale, Jul 03 2024 *) -
Sage
[3^(n-1) -n for n in (2..32)] # G. C. Greubel, Jan 07 2022
Formula
a(n) = 3^(n-1) - n = A061980(n-1, 2). - Henry Bottomley, May 24 2001
From Paul Barry, Jun 24 2003: (Start)
With offset 0, this is 3^(n+1) - n - 2.
Partial sums of A048473. (End)
From Colin Barker, Dec 19 2012: (Start)
a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).
G.f.: x^2*(1 + x)/((1-x)^2*(1-3*x)). (End)
E.g.f.: (exp(3*x) - 3*x*exp(x) - 1)/3. - Wolfdieter Lang, Apr 17 2017
Extensions
More terms from Vladeta Jovovic, Mar 20 2001
Comments
References
Links