A004761 -id:A004761 - cp's OEIS frontend

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

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

A004760 List of numbers whose binary expansion does not begin 10.

Original entry on oeis.org

0, 1, 3, 6, 7, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

N. J. A. Sloane

For n >= 2 sequence {a(n+2)} is the minimal recursive such that A007814(a(n+2))=A007814(n). - Vladimir Shevelev, Apr 27 2009
A053645(a(n)) = n-1 for n > 0. - Reinhard Zumkeller, May 20 2009
a(n+1) is also the number of nodes in a complete binary tree with n nodes in the bottommost level. - Jacob Jona Fahlenkamp, Feb 01 2023

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009. [_Vladimir Shevelev_, Apr 15 2009]

Cf. A004761, A007814, A010060, A159559, A159560, A159615, A159619, A159629, A159698, A004759.

0,1,seq(seq(3*2^d+x,x=0..2^d-1),d=0..6); # Robert Israel, Aug 03 2016

Select[Range@ 125, If[Length@ # < 2, #, Take[#, 2]] &@ IntegerDigits[#, 2] != {1, 0} &] (* Michael De Vlieger, Aug 02 2016 *)

is(n)=n<2 || binary(n)[2] \\ Charles R Greathouse IV, Sep 23 2012

print1("0, 1");for(i=0,5,for(n=3<Charles R Greathouse IV, Sep 23 2012

a(n) = if(n<=2,n-1, (n-=2) + 2<Kevin Ryde, Jul 22 2022

def A004760(n): return m+(1<0 else n-1 # Chai Wah Wu, Jul 26 2023

maxrow <- 8 # by choice
b01 <- 1
for(m in 0:maxrow){
  b01 <- c(b01,rep(1,2^(m+1))); b01[2^(m+1):(2^(m+1)+2^m-1)] <- 0
}
a <- which(b01 == 1)
# Yosu Yurramendi, Mar 30 2017

For n > 0, a(n) = 3n - 2 - A006257(n-1). - Ralf Stephan, Sep 16 2003
a(0) = 0, a(1) = 1, for n > 0: a(2n) = 2*a(n) + 1, a(2n+1) = 2*a(n+1). - Philippe Deléham, Feb 29 2004
For n >= 3, A007814(a(n)) = A007814(n-2). - Vladimir Shevelev, Apr 15 2009
a(n+2) = min{m>a(n+1): A007814(m)=A007814(n)}; A010060(a(n+2)) = 1-A010060(n). - Vladimir Shevelev, Apr 27 2009
a(1)=0, a(2)=1, a(2^m+k+2) = 2^(m+1) + 2^m+k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Jul 30 2016
G.f.: x/(1-x)^2 + (x/(1-x))*Sum_{k>=0} 2^k*x^(2^k). - Robert Israel, Aug 03 2016
a(2^m+k) = A004761(2^m+k) + 2^m, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016
For n > 0, a(n+1) = n + 2^ceiling(log_2(n)) - 1. - Jacob Jona Fahlenkamp, Feb 01 2023

Offset changed to 1, b-file corrected. - N. J. A. Sloane, Aug 07 2016

A004754 Numbers n whose binary expansion starts 10.

Original entry on oeis.org

2, 4, 5, 8, 9, 10, 11, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 128, 129, 130, 131

Offset: 1

N. J. A. Sloane

A000120(a(n)) = A000120(n); A023416(a(n-1)) = A008687(n) for n > 1. - Reinhard Zumkeller, Dec 04 2015

			10 in binary is 1010, so 10 is in sequence.

T. D. Noe, Table of n, a(n) for n = 1..1023
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Index entries for sequences related to binary expansion of n

Cf. A123001 (binary version), A004755 (11), A004756 (100), A004757 (101), A004758 (110), A004759 (111).
Cf. A004760, A053644, A062050, A076877.
Apart from initial terms, same as A004761.
Cf. A000120, A023416, A008687.

import Data.List (transpose)
a004754 n = a004754_list !! (n-1)
a004754_list = 2 : concat (transpose [zs, map (+ 1) zs])
                   where zs = map (* 2) a004754_list
-- Reinhard Zumkeller, Dec 04 2015

w = {1, 0}; Select[Range[2, 131], If[# < 2^(Length@ w - 1), True, Take[IntegerDigits[#, 2], Length@ w] == w] &] (* Michael De Vlieger, Aug 08 2016 *)

PARI
```
a(n)=n+2^floor(log(n)/log(2))
```

is(n)=n>1 && !binary(n)[2] \\ Charles R Greathouse IV, Sep 23 2012

def A004754(n): return n+(1<Chai Wah Wu, Jul 13 2022

a(2n) = 2a(n), a(2n+1) = 2a(n) + 1 + [n==0].
a(n) = n + 2^floor(log_2(n)) = n + A053644(n).
a(2^m+k) = 2^(m+1) + k, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Aug 08 2016

Edited by Ralf Stephan, Oct 12 2003

A326774 For any number m, let m* be the bi-infinite string obtained by repetition of the binary representation of m; this sequence lists the numbers n such that for any k < n, n* does not equal k* up to a shift.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 21, 23, 32, 33, 34, 35, 37, 38, 39, 43, 47, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 146, 147, 149, 150, 151, 154

Offset: 0

Rémy Sigrist, Jul 27 2019

This sequence contains every power of 2.
No term belongs to A121016.
Every terms belongs to A004761.
For any k > 0, there are A001037(k) terms with binary length k.
From Gus Wiseman, Apr 19 2020: (Start)
Also numbers k such that the k-th composition in standard order (row k of A066099) is a co-Lyndon word (regular Lyndon words being A275692). For example, the sequence of all co-Lyndon words begins:
()            37: (3,2,1)         79: (3,1,1,1,1)
(1)           38: (3,1,2)         85: (2,2,2,1)
(2)           39: (3,1,1,1)       87: (2,2,1,1,1)
(3)           43: (2,2,1,1)       91: (2,1,2,1,1)
(2,1)         47: (2,1,1,1,1)     95: (2,1,1,1,1,1)
(4)           64: (7)            128: (8)
(3,1)         65: (6,1)          129: (7,1)
(2,1,1)       66: (5,2)          130: (6,2)
(5)           67: (5,1,1)        131: (6,1,1)
(4,1)         68: (4,3)          132: (5,3)
(3,2)         69: (4,2,1)        133: (5,2,1)
(3,1,1)       70: (4,1,2)        134: (5,1,2)
(2,2,1)       71: (4,1,1,1)      135: (5,1,1,1)
(2,1,1,1)     73: (3,3,1)        137: (4,3,1)
(6)           74: (3,2,2)        138: (4,2,2)
(5,1)         75: (3,2,1,1)      139: (4,2,1,1)
(4,2)         77: (3,1,2,1)      140: (4,1,3)
(4,1,1)       78: (3,1,1,2)      141: (4,1,2,1)
(End)

			3* = ...11... equals 1* = ...1..., so 3 is not a term.
6* = ...110... equals up to a shift 5* = ...101..., so 6 is not a term.
11* = ...1011... only equals up to a shift 13* = ...1101... and 14* = ...1110..., so 11 is a term.

Rémy Sigrist, Logarithmic scatterplot of the first differences of the terms < 2^24
Rémy Sigrist, PARI program for A326774

Cf. A001037, A004761, A065609, A121016.
Necklace compositions are counted by A008965.
Lyndon compositions are counted by A059966.
Length of Lyndon factorization of binary expansion is A211100.
Numbers whose reversed binary expansion is a necklace are A328595.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of Lyndon factorization of reversed binary expansion is A329313.
Length of co-Lyndon factorization of reversed binary expansion is A329326.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774 (this sequence).
- Aperiodic compositions are A328594.
- Reversed co-necklaces are A328595.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
- Reversed necklaces are A333943.
- Length of co-Lyndon factorization is A334029.
Cf. A000740, A034691, A060223, A269134, A292884, A328596.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Select[Range[0,100],colynQ[stc[#]]&] (* Gus Wiseman, Apr 19 2020 *)

PARI
```
See Links section.
```

A171757 Even numbers whose binary expansion begins 10.

Original entry on oeis.org

2, 4, 8, 10, 16, 18, 20, 22, 32, 34, 36, 38, 40, 42, 44, 46, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178

Offset: 1

N. J. A. Sloane, Oct 12 2010

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1024 from R. J. Mathar)

Cf. A004761, A171758, A171763, A171764.

A subsequence of A004754.

n := 1 ;
for k from 2 to 4000 by 2 do
    dgs := convert(k,base,2) ;
    if op(-1,dgs) = 1 and op(-2,dgs) = 0 then
        printf("%d %d\n",n,k) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Jan 31 2015

Select[Range[2, 200, 2], IntegerDigits[#, 2][[1 ;; 2]] == {1, 0} &] (* Amiram Eldar, Sep 01 2020 *)

isok(m) = if (!(m%2), my(b=binary(m)); (b[1]==1) && (b[2]==0)); \\ Michel Marcus, Jun 24 2021

from itertools import count, product, takewhile
def agen(): # generator for sequence
    yield 2
    for digits in count(0):
        for mid in product("01", repeat=digits):
            yield int("10" + "".join(mid) + "0", 2)
def aupto(lim): return list(takewhile(lambda x: x <= lim, agen()))
print(aupto(180)) # Michael S. Branicky, Jun 24 2021

a(n) = 2*A004761(n+1). - Jon Maiga / Georg Fischer, Jun 24 2021

A065629 Permutation of N induced by rotating the node 3 right in the infinite planar binary tree. The third row of A065625. Inverse of A065630.

Original entry on oeis.org

1, 2, 7, 4, 5, 3, 15, 8, 9, 10, 11, 6, 14, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 12, 13, 28, 29, 60, 61, 62, 63, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 25, 26, 27, 56, 57, 58, 59, 120, 121, 122, 123, 124, 125, 126, 127, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Antti Karttunen, Nov 08 2001

nonn

Index entries for sequences that are permutations of the natural numbers

Stabilizes the set A004761.

Maple
```
[seq(RotateNodeRight(3,j),j=1..120)];
```

A065630 Permutation of N induced by rotating the node 3 left in the infinite planar binary tree. The third row of A065626. Inverse of A065629.

Original entry on oeis.org

1, 2, 6, 4, 5, 12, 3, 8, 9, 10, 11, 24, 25, 13, 7, 16, 17, 18, 19, 20, 21, 22, 23, 48, 49, 50, 51, 26, 27, 14, 15, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 96, 97, 98, 99, 100, 101, 102, 103, 52, 53, 54, 55, 28, 29, 30, 31, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92

Offset: 1

Antti Karttunen, Nov 08 2001

nonn

Index entries for sequences that are permutations of the natural numbers

Stabilizes the set A004761.

Maple
```
[seq(RotateNodeLeft(3,j),j=1..120)];
```

A316385 Lexicographically earliest sequence of distinct positive terms such that for any n > 0, a(n) AND a(2*n) = a(n) (where AND denotes the binary AND operator).

Original entry on oeis.org

1, 3, 2, 7, 4, 6, 5, 15, 8, 12, 9, 14, 10, 13, 11, 31, 16, 24, 17, 28, 18, 25, 19, 30, 20, 26, 21, 29, 22, 27, 23, 63, 32, 48, 33, 56, 34, 49, 35, 60, 36, 50, 37, 57, 38, 51, 39, 62, 40, 52, 41, 58, 42, 53, 43, 61, 44, 54, 45, 59, 46, 55, 47, 127, 64, 96, 65

Offset: 1

Rémy Sigrist, Jul 01 2018

This sequence is a permutation of the natural numbers (as odd-indexed terms are not constrained); see A316472 for the inverse sequence.

In the binary plot of the sequence, if the pixel (x, y) is on, then the pixel (2*x, y) is on.

			The first terms, alongside the binary representations of a(n) and of a(2*n), are:
  n  a(n) bin(a(n)) bin(a(2n))
  -- ---- --------- ----------
   1    1         1         11
   2    3        11        111
   3    2        10        110
   4    7       111       1111
   5    4       100       1100
   6    6       110       1110
   7    5       101       1101
   8   15      1111      11111
   9    8      1000      11000
  10   12      1100      11100

Rémy Sigrist, Table of n, a(n) for n = 1..8191
Rémy Sigrist, Colored scatterplot of the first 2^14-1 terms (where the color is function of the 2-adic valuation of n)
Rémy Sigrist, PARI program for A316385
Index entries for sequences that are permutations of the natural numbers

Cf. A004755, A004761, A020989, A316472 (inverse).

PARI
```
See Links section.
```

Empirically:
- a(2*k) = A004755(a(k)) for any k > 0,
- a(2*k - 1) = A004761(k + 1) for any k > 0,
- a(n) = n iff n belongs to A020989.

A337909 Distinct terms of A080079 in the order in which they appear.

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 6, 5, 16, 15, 14, 13, 12, 11, 10, 9, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 128

Offset: 1

Lorenzo Sauras Altuzarra, Sep 29 2020

nonn

This sequence is a permutation of the positive integers.

The cardinality of {2^k, ..., (2^k - 0^k)/2 + 1} is A011782(k).

			(2^0, ..., (2^0 - 0^0)/2 + 1) = (1),
(2^1, ..., (2^1 - 0^1)/2 + 1) = (2),
(2^2, ..., (2^2 - 0^2)/2 + 1) = (4, 3),
(2^3, ..., (2^3 - 0^3)/2 + 1) = (8, 7, 6, 5)...

Index entries for sequences that are permutations of the natural numbers

Cf. A004761, A011782, A053644, A054429, A080079, A280510.

{1}~Join~Array[3*2^(IntegerLength[# - 1, 2] - 1) - # + 1 &, 64, 2] (* Michael De Vlieger, Oct 05 2020 *)

a(n) = if(n--, 3<Kevin Ryde, Sep 29 2020

a(1) = 1 and a(n) = A080079(n - 1 + 2^floor(log_2(n - 1))) if n > 1.
a(n) = A080079(A004761(n+1)).
From Kevin Ryde, Sep 29 2020: (Start)
a(n) = 3*A053644(n-1) - (n-1), if n > 1.
a(n) = A054429(n-1) + 1, if n > 1.
a(n) = A280510(n) - n + 1, if n > 1. (End)

cp's OEIS Frontend

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

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A004760 List of numbers whose binary expansion does not begin 10.

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A004754 Numbers n whose binary expansion starts 10.

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A326774 For any number m, let m* be the bi-infinite string obtained by repetition of the binary representation of m; this sequence lists the numbers n such that for any k < n, n* does not equal k* up to a shift.

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A171757 Even numbers whose binary expansion begins 10.

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A065629 Permutation of N induced by rotating the node 3 right in the infinite planar binary tree. The third row of A065625. Inverse of A065630.

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A065630 Permutation of N induced by rotating the node 3 left in the infinite planar binary tree. The third row of A065626. Inverse of A065629.