A231550 Permutation of nonnegative integers: for each bit[i] in the binary representation, except the most and the least significant bits, set bit[i] = bit[i] XOR bit[i-1], where bit[i-1] is the less significant bit, XOR is the binary logical exclusive or operator.

0, 1, 2, 3, 4, 7, 6, 5, 8, 11, 14, 13, 12, 15, 10, 9, 16, 19, 22, 21, 28, 31, 26, 25, 24, 27, 30, 29, 20, 23, 18, 17, 32, 35, 38, 37, 44, 47, 42, 41, 56, 59, 62, 61, 52, 55, 50, 49, 48, 51, 54, 53, 60, 63, 58, 57, 40, 43, 46, 45, 36, 39, 34, 33, 64, 67, 70, 69, 76

Offset: 0

Author

Alex Ratushnyak, Nov 10 2013

Keywords

• nonn
• look
• base
• easy

Comments

This permutation transforms the enumeration system of positive irreducible fractions A020651/A020650 into the enumeration system A002487/A002487' (Calkin-Wilf), and enumeration system A245327/A245326 into A162911/A162912 (Drib). - Yosu Yurramendi, Jun 16 2015

Links

• Ivan Neretin, Table of n, a(n) for n = 0..8192
• Index entries for sequences that are permutations of the nonnegative integers

Crossrefs

Cf. A003188, A006068, A231551.

Programs

Mathematica

Join[{0, 1}, Table[d = IntegerDigits[n, 2]; FromDigits[Join[{d[[1]]}, BitXor[Most@Rest@d, Rest@Rest@d], {d[[-1]]}], 2], {n, 2, 68}]] (* Ivan Neretin, Dec 28 2016 *)

PARI

a(n) = bitxor(n, if(n>3, bitand(n<<1, bitneg(0,logint(n,2))))); \\ Kevin Ryde, Jul 17 2021

Python

for n in range(99):
  bits = [0]*64
  orig = [0]*64
  l = int.bit_length(int(n))
  t = n
  for i in range(l):
    bits[i] = orig[i] = t&1
    t>>=1
  for i in range(1, l-1):  bits[i] ^= orig[i-1]   # A231550
  #for i in range(1, l-1):  bits[i] ^= bits[i-1]   # A231551
  #for i in range(l-1):  bits[i] ^= orig[i+1]      # A003188
  #for i in range(1,l):  bits[l-1-i] ^= bits[l-i]  # A006068
  t = 0
  for i in range(l):  t += bits[i]<
				

R

a <- 1
maxlevel <- 8 # by choice
#
for(m in 0:maxlevel) for(k in 0:(2^m-1)){
  a[2^(m+1)  +2*k] <- 2*a[2^m+k]
  a[2^(m+2)-1-2*k] <- 2*a[2^m+k] + 1
}
(a <- c(0,a))
# Yosu Yurramendi, Apr 10 2017

Formula

a(A231551(n)) = A231551(a(n)) = n.

a(n) = A284460(A258996(n)) = A092569(A284460(n)), n > 0. - Yosu Yurramendi, Apr 10 2017