A057114 -id:A057114 - cp's OEIS frontend

A065259 A057114 conjugated with A059893, inverse of A065260.

3, 1, 7, 2, 11, 5, 15, 4, 19, 9, 23, 6, 27, 13, 31, 8, 35, 17, 39, 10, 43, 21, 47, 12, 51, 25, 55, 14, 59, 29, 63, 16, 67, 33, 71, 18, 75, 37, 79, 20, 83, 41, 87, 22, 91, 45, 95, 24, 99, 49, 103, 26, 107, 53, 111, 28, 115, 57, 119, 30, 123, 61, 127, 32, 131, 65, 135, 34, 139

Offset: 1

Antti Karttunen, Oct 28 2001

			G.f. = 3*x + x^2 + 7*x^3 + 2*x^4 + 11*x^5 + 5*x^6 + 15*x^7 + 4*x^8 + ...

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Vec(x*(3+x+7*x^2+2*x^3+5*x^4+3*x^5+x^6)/((1-x)^2*(1+x)^2*(1+x^2)^2) + O(x^100)) \\ Colin Barker, Oct 29 2016

{a(n) = if( n%2, 2*n+1, n%4, n-1, n/2)}; /* Michael Somos, Nov 06 2016 */

a(n) = A059893(A057114(A059893(n))).
a(2*k+1) = 4*k+3, a(4*k+2) = 4*k+1, a(4*k+4) = 2*k+2. - Ralf Stephan, Jun 10 2005
a(n) = (11*n+2-(5*n+6)*(-1)^n+(n-2)*(1+(-1)^n)*(-1)^((2*n-3-(-1)^n)/4))/8. - Luce ETIENNE, Oct 29 2016
From Colin Barker, Oct 29 2016: (Start)
a(n) = 2*a(n-4) - a(n-8) for n>8.
G.f.: x*(3 + x + 7*x^2 + 2*x^3 + 5*x^4 + 3*x^5 + x^6)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2). (End)

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

Original entry on oeis.org

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

N. J. A. Sloane

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

			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, ...

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).

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

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

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);

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 *)

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

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

A054424 Permutation of natural numbers: maps the canonical list of fractions (A020652/A020653) to whole Stern-Brocot (Farey) tree (top = 1/1 and both sides < 1 and > 1, but excluding the "fractions" 0/1 and 1/0).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 5, 6, 15, 16, 31, 32, 9, 11, 12, 14, 63, 64, 10, 13, 127, 128, 17, 23, 24, 30, 255, 256, 19, 28, 511, 512, 33, 18, 20, 47, 48, 27, 29, 62, 1023, 1024, 22, 25, 2047, 2048, 65, 35, 39, 21, 95, 96, 26, 56, 60, 126, 4095, 4096, 34, 40, 55, 61, 8191, 8192

Offset: 1

Antti Karttunen

nonn

			Whole Stern-Brocot tree: 1/1 1/2 2/1 1/3 2/3 3/2 3/1 1/4 2/5 3/5 3/4 4/3 5/3 5/2 4/1 1/5 2/7
Canonical fractions: 1/1 1/2 2/1 1/3 3/1 1/4 2/3 3/2 4/1 1/5 5/1 1/6 2/5 3/4 4/3 5/2 6/1

Cf. A047679, A007305, A007306, A054427, A057114. In table form: A054425. Inverse permutation: A054426.

cfrac2binexp := proc(c) local i,e,n; n := 0; for i from 1 to nops(c) do e := c[i]; if(i = nops(c)) then e := e-1; fi; n := ((2^e)*n) + ((i mod 2)*((2^e)-1)); od; RETURN(n); end;
frac2position_in_whole_SB_tree := proc(r) local k,msb; if(1 = r) then RETURN(1); else if(r > 1) then k := cfrac2binexp(convert(r,confrac)); else k := ReflectBinTreePermutation(cfrac2binexp(convert(1/r,confrac))); fi; msb := floor_log_2(k); if(r > 1) then RETURN(k + (2^(msb+1))); else RETURN(k + (2^(msb+1)) - (2^msb)); fi; fi; end;
canonical_fractions_to_whole_SternBrocot_permutation := proc(u) local a,n,i; a := []; for n from 2 to u do for i from 1 to n-1 do if (1 = igcd(n,i)) then a := [op(a),frac2position_in_whole_SB_tree(i/(n-i))]; fi; od; od; RETURN(a); end; # ReflectBinTreePermutation and floor_log_2 given in A054429

canonical_fractions_to_whole_SternBrocot_permutation(30);

A065625 Table of permutations of N, each row being a generator of the "rotation group" of infinite planar binary tree. Inverse generators are given in A065626.

Original entry on oeis.org

3, 1, 1, 7, 5, 1, 2, 3, 2, 1, 6, 2, 7, 2, 1, 14, 11, 4, 3, 2, 1, 15, 6, 5, 9, 3, 2, 1, 4, 7, 3, 5, 4, 3, 2, 1, 5, 4, 15, 6, 11, 4, 3, 2, 1, 12, 10, 8, 7, 6, 5, 4, 3, 2, 1, 13, 22, 9, 4, 7, 13, 5, 4, 3, 2, 1, 28, 23, 10, 19, 8, 7, 6, 5, 4, 3, 2, 1, 29, 12, 11, 10, 9, 8, 15, 6, 5, 4, 3, 2, 1, 30, 13, 6, 11, 5, 9, 8, 7, 6, 5, 4, 3, 2, 1, 31, 14, 14, 12, 23, 10, 9, 17, 7, 6, 5, 4, 3, 2

Offset: 0

Antti Karttunen, Nov 08 2001

Consider the following infinite binary tree, where the nodes are numbered in breadth-first, left-to-right fashion from the top as:
.............................1............................
.............2...............................3............
.....4...............5...............6...............7....
.8.......9.......10.....11.......12.....13.......14.....15
etc., i.e. the node Y is a descendant of the node X, iff its binary expansion (the most significant bits) begin with the binary expansion of X.
In this table the n-th row is a permutation induced by the rotation of the node n right and in the table A065626 the corresponding row gives the inverse of that permutation, induced by rotation of the node n left. Particular realizations of this tree are the Christoffel tree and the Stern-Brocot tree (A007305/A007306), thus each such rotation, or composition of such rotations (e.g. A065249) induces a particular bijective function on rationals and such functions form the "group A" of the order-preserving permutations of the rational numbers as defined by Cameron.

The first row (rotate the top node right): A057114, 2nd row (rotate the top node's left child): A065627, 3rd row (rotate the top node's right child): A065629, 4th row: A065631, 5th row: A065633, 6th row: A065635, 7th row: A065637, 8th row: A065639. Maple procedure floor_log_2 given in A054429, for trinv, follow A065167.

Variant of the same idea: A065658.

[seq(RotateRightTable(j),j=0..119)];
RotateRightTable := n -> RotateNodeRight(1+(n-((trinv(n)*(trinv(n)-1))/2)),(((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1);
# Rewrites t-prefixed x's in the following way: t -> t1, t1... -> t11..., t0 -> t, t01... -> t10..., t00... -> t0... and leaves other x's intact.
RotateNodeRight := proc(t,x) local u,y; u := floor_log_2(t)+1; y := floor_log_2(x)+1; if(y < u) then RETURN(x); fi; if(floor(x/(2^(y-u))) <> t) then RETURN(x); fi; if(x = t) then RETURN((2*x)+1); fi; if(1 = (floor(x/(2^(y-u-1))) mod 2)) then RETURN(x + (t * 2^(y-u)) + 2^(y-u)); fi; if(y = (u+1)) then RETURN(x/2); fi; if(1 = (floor(x/(2^(y-u-2))) mod 2)) then RETURN(x + 2^(y-u-2)); fi; RETURN(x - (t * 2^(y-u-1))); end;

A057115 Order-preserving permutation of the rational numbers (x -> x-1); positions in Stern-Brocot tree.

Original entry on oeis.org

2, 4, 1, 8, 9, 5, 3, 16, 17, 18, 19, 10, 11, 6, 7, 32, 33, 34, 35, 36, 37, 38, 39, 20, 21, 22, 23, 12, 13, 14, 15, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 28, 29, 30, 31, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150

Offset: 1

Antti Karttunen, Aug 09 2000

nonn

Inverse permutation: A057114.
When conjugated with A059893, one gets A065260, a valid siteswap permutation.
The first row of A065626, i.e. a(n) = RotateNodeLeft(1, n).

sbtree_perm_1_1_left := x -> (`if`((x <= 0),x,(`if`((x < 1),(x/(1+x)),(`if`((x < 2),(1/(3-x)),(x-1)))))));

a(n) = frac2position_in_whole_SB_tree(sbtree_perm_1_1_left(SternBrocotTreeNum(n)/SternBrocotTreeDen(n)))

A065249 Permutation of N induced by the order-preserving permutation of the positive rational numbers (x -> x/2), positions in Stern-Brocot tree.

Original entry on oeis.org

2, 8, 1, 32, 4, 11, 6, 128, 16, 35, 18, 5, 47, 24, 3, 512, 64, 131, 66, 17, 143, 72, 9, 21, 44, 23, 191, 96, 12, 27, 14, 2048, 256, 515, 258, 65, 527, 264, 33, 69, 140, 71, 575, 288, 36, 75, 38, 10, 87, 176, 22, 93, 188, 95, 767, 384, 48, 99, 50, 13, 111, 56, 7, 8192

Offset: 1

Antti Karttunen, Oct 25 2001

nonn

Cf. A057114, A065251. Inverse permutation: A065250.

[seq(A065249(j),j=1..120)]; A065249 := n -> frac2position_in_whole_SB_tree((SternBrocotTreeNum(n)/SternBrocotTreeDen(n))/2);

A065263 Infinite binary tree inspired permutation of N: 1 -> 3, 11ab..yz -> 11ab..yz1, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11AB..Y0 (where 1AB..Y0 is the complement of 0ab..y1).

Original entry on oeis.org

3, 1, 7, 2, 6, 13, 15, 4, 14, 5, 12, 25, 27, 29, 31, 8, 30, 9, 28, 10, 26, 11, 24, 49, 51, 53, 55, 57, 59, 61, 63, 16, 62, 17, 60, 18, 58, 19, 56, 20, 54, 21, 52, 22, 50, 23, 48, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 32, 126, 33, 124

Offset: 1

Antti Karttunen, Oct 28 2001

nonn

When an infinite planar binary tree is mapped breadth-first-wise from left to right (1 is at top, 2 is its left and 3 its right child, 4 is 2's left child, etc.) then this permutation induces such rearrangement of its nodes, that on the right side every node replaces its right child, on the left side the left children replace their parents and the right children are reflected to the right side, to be the left children of their new parents.

Index entries for sequences that are permutations of the natural numbers

A057114, A065269, A065275, A065281, A065287. Inverse: A065264, conjugated with A059893: A065265 and the inverse of that: A065266.

RightChildInverted := proc(n) local k; if(1 = n) then RETURN(3); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN((2*n)+1); fi; if(0 = (n mod 2)) then RETURN(n/2); fi; RETURN(2^(k+1) + ((2^(k+2))-1) - n); end;

A065269 Infinite binary tree inspired permutation of N: 1 -> 1, 11ab..yz -> 11ab..yz0, 10ab..y1 -> 10ab..y, 10ab..y0 -> 11AB..Y1 (where 1AB..Y1 is the complement of 0ab..y0).

Original entry on oeis.org

1, 3, 6, 7, 2, 12, 14, 15, 4, 13, 5, 24, 26, 28, 30, 31, 8, 29, 9, 27, 10, 25, 11, 48, 50, 52, 54, 56, 58, 60, 62, 63, 16, 61, 17, 59, 18, 57, 19, 55, 20, 53, 21, 51, 22, 49, 23, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 127, 32, 125, 33

Offset: 1

Antti Karttunen, Oct 28 2001

nonn

On the right side every node replaces its left child, on the left side the right children replace their parents and the left children are reflected to the right side (becoming right children). See comment at A065263.

Index entries for sequences that are permutations of the natural numbers

A057114, A065263, A065275, A065281, A065287. Inverse: A065270, conjugated with A059893: A065271 and the inverse of that: A065272.

LeftChildInverted := proc(n) local k; if(1 = n) then RETURN(1); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN(2*n); fi; if(1 = (n mod 2)) then RETURN((n-1)/2); fi; RETURN(2^(k+1) + ((2^(k+2))-1) - n); end;

A065275 Infinite binary tree inspired permutation of N: 1 -> 3, 11ab..yz -> 11ab..yz0, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11ab..y1.

Original entry on oeis.org

3, 1, 6, 2, 7, 12, 14, 4, 13, 5, 15, 24, 26, 28, 30, 8, 25, 9, 27, 10, 29, 11, 31, 48, 50, 52, 54, 56, 58, 60, 62, 16, 49, 17, 51, 18, 53, 19, 55, 20, 57, 21, 59, 22, 61, 23, 63, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 32, 97, 33, 99

Offset: 1

Antti Karttunen, Oct 28 2001

nonn

On the right side every node replaces its left child, on the left side the left children replace their parents and the right children are transferred to the same offset at the right side (staying right children). See comment at A065263.

Index entries for sequences that are permutations of the natural numbers

A057114, A065263, A065269, A065281, A065287. Inverse: A065276, conjugated with A059893: A065277 and the inverse of that: A065278.

RightChildTransferred := proc(n) local k; if(1 = n) then RETURN(3); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN(2*n); fi; if(0 = (n mod 2)) then RETURN(n/2); fi; RETURN(n + (2^k)); end;

A065281 Infinite binary tree inspired permutation of N: 1 -> 1, 11ab..yz -> 11ab..yz1, 10ab..y1 -> 10ab..y, 10ab..y0 -> 11ab..y0.

Original entry on oeis.org

1, 3, 7, 6, 2, 13, 15, 12, 4, 14, 5, 25, 27, 29, 31, 24, 8, 26, 9, 28, 10, 30, 11, 49, 51, 53, 55, 57, 59, 61, 63, 48, 16, 50, 17, 52, 18, 54, 19, 56, 20, 58, 21, 60, 22, 62, 23, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 96, 32, 98, 33

Offset: 1

Antti Karttunen, Oct 28 2001

nonn

On the right side every node replaces its right child, on the left side the right children replace their parents and the left children are transferred to the same offset at the right side (staying left children). See comment at A065263.

Index entries for sequences that are permutations of the natural numbers

A057114, A065263, A065269, A065275, A065287. Inverse: A065282, conjugated with A059893: A065283 and the inverse of that: A065284.

LeftChildTransferred := proc(n) local k; if(1 = n) then RETURN(1); fi; k := floor_log_2(n)-1; if(3 = floor(n/(2^k))) then RETURN((2*n)+1); fi; if(1 = (n mod 2)) then RETURN((n-1)/2); fi; RETURN(n + (2^k)); end;

cp's OEIS Frontend

A065259 A057114 conjugated with A059893, inverse of A065260.

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

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A054424 Permutation of natural numbers: maps the canonical list of fractions (A020652/A020653) to whole Stern-Brocot (Farey) tree (top = 1/1 and both sides < 1 and > 1, but excluding the "fractions" 0/1 and 1/0).

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A065625 Table of permutations of N, each row being a generator of the "rotation group" of infinite planar binary tree. Inverse generators are given in A065626.

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A057115 Order-preserving permutation of the rational numbers (x -> x-1); positions in Stern-Brocot tree.

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A065249 Permutation of N induced by the order-preserving permutation of the positive rational numbers (x -> x/2), positions in Stern-Brocot tree.

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A065263 Infinite binary tree inspired permutation of N: 1 -> 3, 11ab..yz -> 11ab..yz1, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11AB..Y0 (where 1AB..Y0 is the complement of 0ab..y1).

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Maple

A065269 Infinite binary tree inspired permutation of N: 1 -> 1, 11ab..yz -> 11ab..yz0, 10ab..y1 -> 10ab..y, 10ab..y0 -> 11AB..Y1 (where 1AB..Y1 is the complement of 0ab..y0).

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Maple

A065275 Infinite binary tree inspired permutation of N: 1 -> 3, 11ab..yz -> 11ab..yz0, 10ab..y0 -> 10ab..y, 10ab..y1 -> 11ab..y1.