A332769 Permutation of the positive integers: a(n) = A258996(A054429(n)) = A054429(A258996(n)).

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

Offset

1,2

Comments

Sequence is self-inverse: a(a(n)) = n.

A002487(1+a(n)) = A162911(n) and A002487(a(n)) = A162912(n). So, a(n) generates the enumeration system of positive rationals based on Stern's sequence A002487 called 'drib'.

Given n, one can compute a(n) by taking into account the binary representation of n, and by flipping every second bit starting from the lowest until reaching the highest 1, which is not flipped.

Links

Yosu Yurramendi, Table of n, a(n) for n = 1..8191

Formula

a(A054429(n)) = A054429(a(n)) = A258996(n),

a(A258996(n)) = A258996(a(n)) = A054429(n).

a(n) = A284447(A065190(n)) = A065190(A284447(n)),

a(A065190(n)) = A065190(a(n)) = A284447(n),

a(A284447(n)) = A284447(a(n)) = A065190(n).

a(A231551(n)) = A154437(n), a(A154437(n)) = A231551(n).

a(A153154(n)) = A284459(n), a(A284459(n)) = A153154(n).

a(1) = 1, a(2) = 3, a(3) = 2; for n > 3, a(2*n) = 2*a(A054429(n)) + 1, a(2*n+1) = 2*a(A054429(n)).

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

Example

n = 23 =  10111_2
            x x
          10010_2 = 18 = a(n).
n = 33 = 100001_2
          x x x
         110100_2 = 52 = a(n).

Prog

(R)
maxrow <- 6 # by choice
a <- 1
for(m in 0:maxrow) for(k in 0:(2^m-1)){
a[2^(m+1)+2*k  ] <- 2*a[2^(m+1)-1-k] + 1
a[2^(m+1)+2*k+1] <- 2*a[2^(m+1)-1-k]
}
a
(R) # Given n, compute a(n) by taking into account the binary representation of n
maxblock <- 7 # 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
  anbit[seq(1, length(anbit) - 1, 2)] <- 1 - anbit[seq(1, length(anbit) - 1, 2)]
  a <- c(a, sum(anbit*2^(0:(length(anbit) - 1))))
}
a
# Yosu Yurramendi, Mar 30 2021
(PARI) a(n) = bitxor(n, 2<<bitor(logint(n, 2)-1, 1)\3); \\ Kevin Ryde, Mar 30 2021

Crossrefs

Similar R-programs: A258996, A284447.

Sequence in context: A080261 A138310 A270197 * A191709 A194077 A257334

Adjacent sequences: A332766 A332767 A332768 * A332770 A332771 A332772

Keyword

nonn

Author

Yosu Yurramendi, Feb 23 2020

Status

approved