A054429 Simple self-inverse permutation of natural numbers: List each block of 2^n numbers (from 2^n to 2^(n+1) - 1) in reverse order.

1, 3, 2, 7, 6, 5, 4, 15, 14, 13, 12, 11, 10, 9, 8, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 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, 32, 127, 126, 125, 124, 123, 122, 121

Offset: 1

Author

Antti Karttunen

Keywords

• nonn
• easy
• look

Comments

a(n) gives the position of the inverse of the n-th term in the full Stern-Brocot tree: A007305(a(n)+2) = A047679(n) and A047679(a(n)) = A007305(n+2). - Reinhard Zumkeller, Dec 22 2008

From Gary W. Adamson, Jun 21 2012: (Start)

The mapping and conversion rules are as follows:

By rows, we have ...

1;

3, 2;

7, 6, 5, 4;

15, 14, 13, 12, 11, 10, 9, 8;

... onto which we are to map one-half of the Stern-Brocot infinite Farey Tree:

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

...

The conversion rules are: Convert the decimal to binary, adding a duplicate of the rightmost binary term to its right. For example, 10 = 1010, which becomes 10100. Then, from the left, record the number of runs = [1,1,1,2], the continued fraction representation of 5/8. Check: 10 decimal corresponds to 5/8 as shown in the overlaid mapping. Take decimal 9 = 1001 which becomes 10011, with a continued fraction representation of [1,2,2] = 5/7. Check: 9 decimal corresponds to 5/7 in the Farey Tree map. (End)

From Indranil Ghosh, Jan 19 2017: (Start)

a(n) is the value generated when n is converted into its Elias gamma code, the 1's and 0's are interchanged and the resultant is converted back to its decimal value for all values of n > 1. For n = 1, A054429(n) = 1 but after converting 1 to Elias gamma code, interchanging the 1's and 0's and converting it back to decimal, the result produced is 0.

For example, let n = 10. The Elias gamma code for 10 is '1110010'. After interchanging the 1's and 0's it becomes "0001101" and 1101_2 = 13_10. So a(10) = 13. (End)

From Yosu Yurramendi, Mar 09 2017 (similar to Zumkeller's comment): (Start)

A002487(a(n)) = A002487(n+1), A002487(a(n)+1) = A002487(n), n > 0.

A162909(a(n)) = A162910(n), A162910(a(n)) = A162909(n), n > 0.

A162911(a(n)) = A162912(n), A162912(a(n)) = A162911(n), n > 0.

A071766(a(n)) = A245326(n), A245326(a(n)) = A071766(n), n > 0.

A229742(a(n)) = A245325(n), A245325(a(n)) = A229742(n), n > 0.

A020651(a(n)) = A245327(n), A245327(a(n)) = A020651(n), n > 0.

A020650(a(n)) = A245328(n), A245328(a(n)) = A020650(n), n > 0. (End)

From Yosu Yurramendi, Mar 29 2017: (Start)

A063946(a(n)) = a(A063946(n)) = A117120(n), n > 0.

A065190(a(n)) = a(A065190(n)) = A092569(n), n > 0.

A258746(a(n)) = a(A258746(n)) = A165199(n), n > 0.

A258996(a(n)) = a(A258996(n)), n > 0.

A117120(a(n)) = a(A117120(n)), n > 0.

A092569(a(n)) = a(A092569(n)), n > 0. (End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Ralf Stephan, Some divide-and-conquer sequences ...
• Ralf Stephan, Table of generating functions
• Index entries for sequences that are permutations of the natural numbers
• Index entries for sequences related to binary expansion of n

Crossrefs

See also A054424, A054430.

{A000027, A054429, A059893, A059894} form a 4-group.

Cf. A053644, A115303, A115304, A115305, A115306, A115307, A115308, A115309, A106649.

This is Guy Steele's sequence GS(6, 5) (see A135416).

Programs

Haskell

a054429 n = a054429_list !! (n-1)
a054429_list = f [1..] where
   f xs@(x:_) = reverse us ++ f vs where (us, vs) = splitAt x xs
-- Reinhard Zumkeller, Jun 01 2015, Feb 21 2014

Maple

A054429 := n -> 3*2^ilog2(n) - n - 1:
seq(A054429(n), n = 1..70); # [Updated by Peter Luschny, Apr 24 2024]

Mathematica

Flatten[Table[Range[2^(n+1)-1,2^n,-1],{n,0,6}]] (* Harvey P. Dale, Dec 17 2013 *)

PARI

A054429(n)= 3<<#binary(n\2)-n-1 \\ M. F. Hasler, Aug 18 2014

Python

from itertools import count, islice
def A054429_gen(): # generator of terms
    return (m for n in count(0) for m in range((1<A054429_list = list(islice(A054429_gen(),30)) # Chai Wah Wu, Jul 27 2023

R

maxblock <- 10 # by choice
a <- NULL
for(m in 0:maxblock) a <- c(a, rev(2^m:(2^(m+1)-1)))
a
# Yosu Yurramendi, Mar 10 2017

Formula

a(n) = ReflectBinTreePermutation(n).

a(n) = if n=1 then 1 else 2*a(floor(n/2)) + 1 - n mod 2. - Reinhard Zumkeller, Feb 18 2003

G.f.: 1/(1-x) * ((x-2x^2)/(1-x) + Sum_{k>=0} 3*2^k*x^2^k). - Ralf Stephan, Sep 15 2003

A000120(a(n)) = A000120(A059894(n)) = A023416(n) + 1. - Ralf Stephan, Oct 05 2003

A115310(n, 1) = a(n). - Reinhard Zumkeller, Jan 20 2006

a(1) = 1, a(2^(m+1) + k) = a(2^m+k) + 2^(m+1),

a(2^(m+1) + 2^m+k) = a(2^m+k) + 2^m, m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Apr 06 2017

a(n) = A117120(A063946(n)) = A063946(A117120(n)) = A092569(A065190(n)) = A065190(A092569(n)), n > 0. - Yosu Yurramendi, Apr 10 2017

a(n) = 3*A053644(n) - n - 1. - Alan Michael Gómez Calderón, Feb 28 2025