A153153 -id:A153153 - cp's OEIS frontend

A153154 Permutation of natural numbers: A059893-conjugate of A006068.

0, 1, 3, 2, 7, 4, 5, 6, 15, 8, 9, 14, 11, 12, 13, 10, 31, 16, 17, 30, 19, 28, 29, 18, 23, 24, 25, 22, 27, 20, 21, 26, 63, 32, 33, 62, 35, 60, 61, 34, 39, 56, 57, 38, 59, 36, 37, 58, 47, 48, 49, 46, 51, 44, 45, 50, 55, 40, 41, 54, 43, 52, 53, 42, 127, 64, 65, 126, 67, 124

Offset: 0

Antti Karttunen, Dec 20 2008

A002487(1+a(n)) = A020651(n) and A002487(a(n)) = A020650(n). So, it generates the enumeration system of positive rationals based on Stern's sequence A002487. - Yosu Yurramendi, Feb 26 2020

Inverse: A153153. a(n) = A059893(A006068(A059893(n))).

maxn <- 63 # by choice
a <- c(1,3,2)
#
for(n in 2:maxn){
  a[2*n] <- 2*a[n] + 1
  if(n%%2==0) a[2*n+1] <- 2*a[n+1]
  else        a[2*n+1] <- 2*a[n-1]
}
(a <- c(0,a))
# Yosu Yurramendi, Feb 26 2020

# Given n, compute a(n) by taking into account the binary representation of n
maxblock <- 8 # by choice
a <- c(1, 3, 2)
for(n in 4:2^maxblock){
  ones <- which(as.integer(intToBits(n)) == 1)
  nbit <- as.integer(intToBits(n))[1:tail(ones, n = 1)]
  anbit <- nbit
  for(i in 2:(length(anbit) - 1))
    anbit[i] <- bitwXor(anbit[i], anbit[i - 1])  # ?bitwXor
  anbit[0:(length(anbit) - 1)] <- 1 - anbit[0:(length(anbit) - 1)]
  a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
(a <- c(0, a))
# Yosu Yurramendi, Oct 04 2021

From Yosu Yurramendi, Feb 26 2020: (Start)
a(1) = 1, for all n > 0 a(2*n) = 2*a(n) + 1, a(2*n+1) = 2*a(A065190(n)).
a(1) = 1, a(2) = 3, a(3) = 2, for all n > 1 a(2*n) = 2*a(n) + 1, and if n even a(2*n+1) = 2*a(n+1), else a(2*n+1) = 2*a(n-1).
a(n) = A054429(A231551(n)) = A231551(A065190(n)) = A284459(A054429(n)) =
       A332769(A284459(n)) = A258996(A154437(n)). (End)

A154438 Permutation of nonnegative integers: A059893-conjugate of A154436.

Original entry on oeis.org

0, 1, 3, 2, 7, 4, 5, 6, 15, 12, 9, 10, 11, 8, 13, 14, 31, 28, 25, 26, 19, 16, 21, 22, 23, 20, 17, 18, 27, 24, 29, 30, 63, 60, 57, 58, 51, 48, 53, 54, 39, 36, 33, 34, 43, 40, 45, 46, 47, 44, 41, 42, 35, 32, 37, 38, 55, 52, 49, 50, 59, 56, 61, 62, 127, 124, 121, 122, 115, 112

Offset: 0

Antti Karttunen, Jan 17 2009

This permutation is induced by the same Lamplighter group generating wreath recursion (binary transducer) as A154436, starting from the active (swapping) state a, but in contrast to it, this one rewrites the bits from the least significant end up to the second most significant bit.

Inverse: A154437.

a(n) = A059893(A154436(A059893(n))) = A054429(A153153(A054429(n))).

maxlevel <- 5 # by choice
a <- 1
for(m in 0:maxlevel) for(k in 0:(2^m-1)){
  a[2^(m+2)-1-2*k] <- 2*a[2^m+k]
  a[2^(m+1)  +2*k] <- 2*a[2^m+k] + 1
}
(a <- c(0,a))
# Yosu Yurramendi, Apr 10 2020

a(0) = 0, a(1) = 1, m > 0, 0 <= k < 2^m a(2^(m+2)-1-2*k) = 2*a(2^m+k),

a(2^(m+1)+2*k) = 2*a(2^m+k) + 1. - Yosu Yurramendi, Apr 10 2020

cp's OEIS Frontend

A153154 Permutation of natural numbers: A059893-conjugate of A006068.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

R

R

Formula