A356331 -id:A356331 - cp's OEIS frontend

A057889 Bijective bit-reverse of n: keep the trailing zeros in the binary expansion of n fixed, but reverse all the digits up to that point.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 25, 20, 21, 26, 29, 24, 19, 22, 27, 28, 23, 30, 31, 32, 33, 34, 49, 36, 41, 50, 57, 40, 37, 42, 53, 52, 45, 58, 61, 48, 35, 38, 51, 44, 43, 54, 59, 56, 39, 46, 55, 60, 47, 62, 63, 64, 65, 66, 97, 68, 81, 98, 113

Offset: 0

Marc LeBrun, Sep 25 2000

The original name was "Bit-reverse of n, including as many leading as trailing zeros." - Antti Karttunen, Dec 25 2024
A permutation of integers consisting only of fixed points and pairs. a(n)=n when n is a binary palindrome (including as many leading as trailing zeros), otherwise a(n)=A003010(n) (i.e. n has no axis of symmetry). A057890 gives the palindromes (fixed points, akin to A006995) while A057891 gives the "antidromes" (pairs). See also A280505.
This is multiplicative in domain GF(2)[X], i.e. with carryless binary arithmetic. A193231 is another such permutation of natural numbers. - Antti Karttunen, Dec 25 2024

			a(6)=6 because 0110 is a palindrome, but a(11)=13 because 1011 reverses into 1101.

N. J. A. Sloane, Table of n, a(n) for n = 0..16384, May 30 2016 [First 8192 terms from _Ivan Neretin_, Jul 09 2015]
Index entries for sequences related to binary expansion of n
Index entries for sequences related to polynomials in ring GF(2)[X]
Index entries for sequences that are permutations of the natural numbers

Cf. A030101, A000265, A006519, A006995, A057890, A057891, A280505, A280508, A331166 [= min(n,a(n))], A366378 [k for which a(k) = k (mod 3)], A369044 [= A014963(a(n))].
Similar permutations for other bases: A263273 (base-3), A264994 (base-4), A264995 (base-5), A264979  (base-9).
Other related (binary) permutations: A056539, A193231.
Compositions of this permutation with other binary (or other base-related) permutations: A264965, A264966, A265329, A265369, A379471, A379472.
Compositions with permutations involving prime factorization: A245450, A245453, A266402, A266404, A293448, A366275, A366276.
Other derived permutations: A246200 [= a(3*n)/3], A266351, A302027, A302028, A345201, A356331, A356332, A356759, A366389.
See also A235027 (which is not a permutation).

Table[FromDigits[Reverse[IntegerDigits[n, 2]], 2]*2^IntegerExponent[n, 2], {n, 71}] (* Ivan Neretin, Jul 09 2015 *)

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2))); \\ Antti Karttunen, Dec 25 2024

def a(n):
    x = bin(n)[2:]
    y = x[::-1]
    return int(str(int(y))+(len(x) - len(str(int(y))))*'0', 2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 11 2017

def A057889(n): return int(bin(n>>(m:=(~n&n-1).bit_length()))[-1:1:-1],2)<Chai Wah Wu, Dec 25 2024

a(n) = A030101(A000265(n)) * A006519(n), with a(0)=0.

Clarified the name with May 30 2016 comment from N. J. A. Sloane, and moved the old name to the comments - Antti Karttunen, Dec 25 2024

A356332 Bit-reverse the odd part of the negaFibonacci representation of -n (and negate): a(n) = -A356327(A057889(A215025(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 6, 7, 8, 9, 5, 11, 12, 31, 27, 23, 16, 17, 28, 19, 20, 21, 22, 15, 24, 30, 26, 14, 18, 29, 25, 13, 32, 33, 86, 82, 65, 71, 38, 78, 61, 57, 42, 51, 44, 45, 72, 83, 74, 62, 50, 43, 75, 53, 54, 55, 56, 41, 58, 77, 70, 40, 49, 63, 64, 36, 79, 85

Offset: 0

Rémy Sigrist, Aug 04 2022

This sequence is a self-inverse permutation of the nonnegative integers similar to A343150, A344682, A345201 and A356331.

			The first terms, alongside the corresponding negaFibonacci representations, are:
  n   a(n)  nega(-n)  nega(-a(n))
  --  ----  --------  -----------
   0     0         0            0
   1     1        10           10
   2     2      1001         1001
   3     3      1000         1000
   4     4      1010         1010
   5    10    100101       101001
   6     6    100100       100100
   7     7    100001       100001
   8     8    100000       100000
   9     9    100010       100010
  10     5    101001       100101
  11    11    101000       101000
  12    12    101010       101010
  13    31  10010101     10101001

Rémy Sigrist, Table of n, a(n) for n = 0..10945
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A057889, A215025, A343150, A344682, A345201, A356327, A356331.

PARI
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See Links section.
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a(a(n)) = n.

a(n) < A000045(2*k+1) iff n < A000045(2*k+1).

A356759 Bit-reverse the odd part of the dual Zeckendorf representation of n: a(n) = A022290(A057889(A003754(n+1))).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 15, 17, 13, 16, 14, 18, 19, 20, 25, 22, 28, 30, 21, 26, 29, 23, 27, 24, 31, 32, 33, 41, 46, 36, 43, 38, 49, 51, 34, 42, 37, 47, 50, 35, 44, 48, 39, 45, 40, 52, 53, 54, 67, 59, 75, 80, 56, 70, 77, 62, 72, 64, 83, 85, 55

Offset: 0

Rémy Sigrist, Aug 26 2022

This sequence is a self-inverse permutation of the nonnegative integers, similar to A345201 and A356331.

The dual Zeckendorf (or lazy Fibonacci) representation expresses uniquely a number n as a sum of distinct positive Fibonacci numbers; these distinct Fibonacci numbers can be encoded in binary, and the corresponding binary encoding, A003754(n+1), cannot have two consecutive nonleading 0's.

			For n = 49:
- the dual Zeckendorf representation of 49 is "1111010",
- reversing its odd part ("111101"), we obtain "1011110",
- so a(49) = 39.

Rémy Sigrist, Table of n, a(n) for n = 0..10945
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A000045, A003714, A003754, A022290, A057889, A104326, A345201, A356331.

PARI
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See Links section.
```

a(a(n)) = n.

a(n) < A000045(k) iff n < A000045(k).

cp's OEIS Frontend

A057889 Bijective bit-reverse of n: keep the trailing zeros in the binary expansion of n fixed, but reverse all the digits up to that point.

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A356332 Bit-reverse the odd part of the negaFibonacci representation of -n (and negate): a(n) = -A356327(A057889(A215025(n))).

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A356759 Bit-reverse the odd part of the dual Zeckendorf representation of n: a(n) = A022290(A057889(A003754(n+1))).

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