A284459 -id:A284459 - cp's OEIS frontend

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A245327 Numerators in recursive bijection from positive integers to positive rationals, where the bijection is f(1) = 1, f(2n) = 1/(f(n)+1), f(2n+1) = f(n)+1.

1, 1, 2, 2, 3, 1, 3, 3, 5, 2, 5, 3, 4, 1, 4, 5, 8, 3, 8, 5, 7, 2, 7, 4, 7, 3, 7, 4, 5, 1, 5, 8, 13, 5, 13, 8, 11, 3, 11, 7, 12, 5, 12, 7, 9, 2, 9, 7, 11, 4, 11, 7, 10, 3, 10, 5, 9, 4, 9, 5, 6, 1, 6, 13, 21, 8, 21, 13, 18, 5, 18, 11, 19, 8, 19, 11, 14, 3, 14, 12, 19, 7, 19, 12, 17, 5, 17, 9, 16, 7, 16, 9, 11, 2, 11, 11, 18, 7, 18, 11

Offset: 1

Yosu Yurramendi, Jul 18 2014

a(n)/A245328(n) enumerates all the reduced nonnegative rational numbers exactly once.
If the terms (n>0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...
   1,
   1, 2,
   2, 3,1, 3,
   3, 5,2, 5,3, 4,1, 4,
   5, 8,3, 8,5, 7,2, 7,4, 7,3, 7,4,5,1,5,
   8,13,5,13,8,11,3,11,7,12,5,12,7,9,2,9,7,11,4,11,7,10,3,10,5,9,4,9,5,6,1,6,
then the sum of the m-th row is 3^m (m = 0,1,2,), and each column k is a Fibonacci sequence.
If the rows are written in a right-aligned fashion:
                                                                        1,
                                                                      1,2,
                                                                  2,3,1,3,
                                                          3,5,2,5,3,4,1,4,
                                      5, 8,3, 8,5, 7,2, 7,4,7,3,7,4,5,1,5,
8,13,5,13,8,11,3,11,7,12,5,12,7,9,2,9,7,11,4,11,7,10,3,10,5,9,4,9,5,6,1,6,
then each column is an arithmetic sequence.
If the sequence is considered by blocks of length 2^m, m = 0,1,2,..., the blocks of this sequence are permutations of terms of blocks from A002487 (Stern's diatomic series or Stern-Brocot sequence), and, more precisely, the reverses of blocks of A020650 ( a(2^m+k) = A020650(2^(m+1)-1-k), m = 0,1,2,..., k = 0,1,2,...,2^m-1).
Moreover, each block is the bit-reversed permutation of the corresponding block of A245325.

Michael De Vlieger, Table of n, a(n) for n = 1..16383, rows 1-14, flattened.
Index entries for fraction trees

Cf. A002487, A020651, A245325, A245328, A273493.

f[n_] := Which[n == 1, 1, EvenQ@ n, 1/(f[n/2] + 1), True, f[(n - 1)/2] + 1]; Table[Numerator@ f@ k, {n, 7}, {k, 2^(n - 1), 2^n - 1}] // Flatten (* Michael De Vlieger, Mar 02 2017 *)

a(n) = my(A=0); forstep(i=logint(n, 2), 0, -1, if(bittest(n, i), A++, A=1/(A+1))); numerator(A) \\ Mikhail Kurkov, Mar 12 2023

N  <- 25 # arbitrary
a <- c(1,1,2)
for(n in 1:N){
  a[4*n]   <-          a[2*n+1]
  a[4*n+1] <- a[2*n] + a[2*n+1]
  a[4*n+2] <- a[2*n]
  a[4*n+3] <- a[2*n] + a[2*n+1]
}
a

a(2n) = A245328(2n+1) , a(2n+1) = A245328(2n) , n=0,1,2,3,...
a((2*n+1)*2^m - 2) = A273493(n), n > 0, m > 0. For n = 0, m > 0, A273493(0) = 1 is needed. For n = 1, m = 0, A273493(0) = 1 is needed. For n > 1, m = 0, numerator((2*n-1) = num+den(n-1). - Yosu Yurramendi, Mar 02 2017
a(n) = A002487(A284459(n)). - Yosu Yurramendi, Aug 23 2021

A231551 Position of n in A231550.

Original entry on oeis.org

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

Offset: 0

Alex Ratushnyak, Nov 10 2013

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

Cf. A003188, A006068, A231550.

Join[{0, 1}, Table[d = Reverse@IntegerDigits[n, 2]; FromDigits[Reverse@Append[FoldList[BitXor, d[[1]], Most@Rest@d], d[[-1]]], 2], {n, 2, 67}]] (* Ivan Neretin, Dec 28 2016 *)

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

maxrow <- 8 # by choice b01 <- 0 # b01 is going to be A010059 a <- 1 for(m in 0:maxrow) for(k in 0:(2^m-1)){ b01[2^(m+1)+ k] <- b01[2^m+k] a[2^(m+1)+ k] <- a[2^m+k] + 2^(m+b01[2^(m+1)+ k]) b01[2^(m+1)+2^m+k] <- 1 - b01[2^m+k] a[2^(m+1)+2^m+k] <- a[2^m+k] + 2^(m+b01[2^(m+1)+2^m+k]) } (a <- c(0,a)) # Yosu Yurramendi, Apr 10 2017

R

maxblock <- 8 # by choice a <- 1:3 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 a <- c(a, sum(anbit*2^(0:(length(anbit) - 1)))) } (a <- c(0,a)) # Yosu Yurramendi, Apr 25 2021

Formula

A231550(a(n)) = a(A231550(n)) = n.

a(n) = A258996(A284460(n)) = A284459(A092569(n)), n > 0. - Yosu Yurramendi, Apr 10 2017

a(n) = A054429(A153154(n)), n > 0. - Yosu Yurramendi, Oct 04 2021

A245328 Denominators in recursive bijection from positive integers to positive rationals (the bijection is f(1) = 1, f(2n) = 1/(f(n)+1), f(2n+1) = f(n)+1).

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 1, 5, 3, 5, 2, 4, 3, 4, 1, 8, 5, 8, 3, 7, 5, 7, 2, 7, 4, 7, 3, 5, 4, 5, 1, 13, 8, 13, 5, 11, 8, 11, 3, 12, 7, 12, 5, 9, 7, 9, 2, 11, 7, 11, 4, 10, 7, 10, 3, 9, 5, 9, 4, 6, 5, 6, 1, 21, 13, 21, 8, 18, 13, 18, 5, 19, 11, 19, 8, 14, 11, 14, 3, 19, 12, 19, 7, 17, 12, 17, 5, 16, 9, 16, 7, 11, 9, 11, 2, 18, 11, 18, 7, 15

Offset: 1

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Author

Yosu Yurramendi, Jul 18 2014

Keywords

nonn
frac

Comments

A245327(n)/a(n) enumerates all the reduced nonnegative rational numbers exactly once.

If the terms (n>0) are written as an array (left-aligned fashion) with rows of length 2^m, m = 0,1,2,3,...

1,

2,1,

3,2, 3,1,

5,3, 5,2, 4,3, 4,1,

8,5, 8,3, 7,5, 7,2, 7,4, 7,3,5,4,5,1,

13,8,13,5,11,8,11,3,12,7,12,5,9,7,9,2,11,7,11,4,10,7,10,3,9,5,9,4,6,5,6,1,

then the sum of the m-th row is 3^m (m = 0,1,2,), and each column k is a Fibonacci sequence.

If the rows are written in a right-aligned fashion:

1,

2,1,

3,2,3,1,

5,3,5,2,4,3,4,1,

8,5, 8,3, 7,5, 7,2,7,4,7,3,5,4,5,1,

13,8,13,5,11,8,11,3,12,7,12,5,9,7,9,2,11,7,11,4,10,7,10,3,9,5,9,4,6,5,6,1,

then each column is an arithmetic sequence. The differences of the arithmetic sequences, except the first on the right, give the sequence A093873 (Numerators in Kepler's tree of harmonic fractions) (a(2^(m+1)-1-k) - a(2^m-1-k) = A093873(k), m = 0,1,2,..., k = 0,1,2,...,2^m-1).

If the sequence is considered by blocks of length 2^m, m = 0,1,2,..., the blocks of this sequence are permutations of terms of blocks from A002487 (Stern's diatomic series or Stern-Brocot sequence), and, more precisely, the reverses of blocks of A020651 ( a(2^m+k) = A020651(2^(m+1)-1-k), m = 0,1,2,..., k = 0,1,2,...,2^m-1).

Moreover, each block is the bit-reversed permutation of the corresponding block of A245326.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..16383, rows 1-14, flattened.

Index entries for fraction trees

Crossrefs

Cf. A002487, A020651, A093873, A245326, A245327, A273493.

Programs

Mathematica

f[n_] := Which[n == 1, 1, EvenQ@ n, 1/(f[n/2] + 1), True, f[(n - 1)/2] + 1]; Table[Denominator@ f@ k, {n, 7}, {k, 2^(n - 1), 2^n - 1}] // Flatten (* Michael De Vlieger, Mar 02 2017 *)

PARI

a(n) = my(A=0); forstep(i=logint(n, 2), 0, -1, if(bittest(n, i), A++, A=1/(A+1))); denominator(A) \\ Mikhail Kurkov, Mar 12 2023

R

N <- 25 # arbitrary a <- c(1,2,1) for(n in 1:N){ a[4*n] <- a[2*n] + a[2*n+1] a[4*n+1] <- a[2*n] a[4*n+2] <- a[2*n] + a[2*n+1] a[4*n+3] <- a[2*n+1] } a

Formula

a(2n) = A245327(2n+1) , a(2n+1) = A245328(2n) , n=1,2,3,...

a((2*n+1)*2^m - 1) = A273493(n), n > 0, m >= 0. For n = 0 A273493(0) = 1 is needed. - Yosu Yurramendi, Mar 02 2017

a(n) = A002487(1+A284459(n)). - Yosu Yurramendi, Aug 23 2021

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

Original entry on oeis.org

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

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Author

Antti Karttunen, Dec 20 2008

Keywords

nonn
base

Comments

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

Links

A. Karttunen, Table of n, a(n) for n = 0..2047

Index entries for sequences that are permutations of the natural numbers

Crossrefs

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

Programs

R

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

R

# 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

Formula

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)

A154437 Permutation of nonnegative integers: A059893-conjugate of A154435.

Original entry on oeis.org

0, 1, 3, 2, 5, 6, 7, 4, 13, 10, 11, 12, 9, 14, 15, 8, 21, 26, 27, 20, 25, 22, 23, 24, 29, 18, 19, 28, 17, 30, 31, 16, 53, 42, 43, 52, 41, 54, 55, 40, 45, 50, 51, 44, 49, 46, 47, 48, 37, 58, 59, 36, 57, 38, 39, 56, 61, 34, 35, 60, 33, 62, 63, 32, 85, 106, 107, 84, 105, 86, 87

Offset: 0

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Author

Antti Karttunen, Jan 17 2009

Keywords

nonn
base

Comments

This permutation is induced by the same Lamplighter group generating wreath recursion (binary transducer) as A154435, 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.

Links

A. Karttunen, Table of n, a(n) for n = 0..2047

Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A154438. a(n) = A059893(A154435(A059893(n))) = A054429(A153154(A054429(n))).

Programs

R

maxn <- 63 # by choice a <- c(1,3,2) for(n in 2:maxn){ a[2*n+1] <- 2*a[n] if(n%%2 == 0) a[2*n] <- 2*a[n+1] + 1 else a[2*n] <- 2*a[n-1] + 1 } (a <- c(0,a)) # Yosu Yurramendi, Feb 23 2020

Formula

From Yosu Yurramendi, Feb 23 2020: (Start)

a(n) = A054429(A284459(n)) = A258996(A153154(n)) = A284459(A065190(n)).

a(1) = 1; for n > 0, a(2*n) = 2*a(A065190(n)) + 1, a(2*n+1) = 2*a(n). (End)

A273493 a(n) = A245327(n) + A245328(n).

Original entry on oeis.org

2, 3, 3, 5, 5, 4, 4, 8, 8, 7, 7, 7, 7, 5, 5, 13, 13, 11, 11, 12, 12, 9, 9, 11, 11, 10, 10, 9, 9, 6, 6, 21, 21, 18, 18, 19, 19, 14, 14, 19, 19, 17, 17, 16, 16, 11, 11, 18, 18, 15, 15, 17, 17, 13, 13, 14, 14, 13, 13, 11, 11, 7, 7, 34, 34, 29, 29, 31, 31, 23, 23, 30, 30, 27, 27, 25, 25, 17, 17, 31, 31, 26, 26, 29, 29, 22, 22, 25

Offset: 1

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Author

Yosu Yurramendi, May 23 2016

Keywords

nonn

Comments

The terms (n>0) may be written as a left-justified array with rows of length 2^m, m >= 0:

2,

3, 3,

5, 5, 4, 4,

8, 8, 7, 7, 7, 7, 5, 5,

13,13,11,11,12,12, 9, 9,11,11,10,10, 9, 9, 6, 6,

21,21,18,18,19,19,14,14,19,19,17,17,16,16,11,11,18,18,15,15,17,17,13,13,14,14,...

All columns have the Fibonacci sequence property: a(2^(m+2) + k) = a(2^(m+1) + k) + a(2^m + k), m >= 0, 0 <= k < 2^m (empirical observations).

The terms (n>0) may also be written as a right-justified array with rows of length 2^m, m >= 0:

2,

3, 3,

5, 5, 4, 4,

8, 8, 7, 7, 7, 7, 5, 5,

13,13,11,11,12,12, 9, 9,11,11,10,10, 9, 9, 6, 6,

...,19,19,17,17,16,16,11,11,18,18,15,15,17,17,13,13,14,14,13,13,11,11, 7, 7,

Each column is an arithmetic sequence. The differences of the arithmetic sequences repeat the sequence A071585: a(2^(m+2) -1 - 2k) - a(2^(m+1) -1 - 2k) = A071585(k-1), m > 0, 0 <= k < 2^m ; a(2^(m+2) -1 - 2k - 1) - a(2^(m+1) -1 - 2k - 1) = A071585(k-1), m > 0, 0 <= k < 2^m .

n>1 occurs in this sequence phi(n) = A000010(n) times, as it occurs in A007306 (Franklin T. Adams-Watters' comment), that is the sequence obtained by adding numerator and denominator in the Calkin-Wilf enumeration system of positive rationals. A245327(n)/A245328(n) is also an enumeration system of all positive rationals, and in each level m >= 0 (ranks between 2^m and 2^(m+1)-1) rationals are the same in both systems. Thus a(n) has the same terms in each level as A007306.

a(n) = A273494(A059893(n)), a(A059893(n)) = A273494(n), n > 0. - Yosu Yurramendi, May 30 2017

Links

Yosu Yurramendi, Table of n, a(n) for n = 1..4099

Crossrefs

Cf. A007306, A268087, A071585, A086592, A273494

Programs

PARI

b(n) = my(b=binary(n)); fromdigits(concat(b[1], Vecrev(vector(#b-1, k, b[k+1]))), 2); \\ from A059893 a(n) = my(n=b(n), x=1, y=1); for(i=0, logint(n, 2), if(bittest(n, i), [x, y]=[x+y, y], [x, y]=[y, x+y])); x \\ Mikhail Kurkov, Mar 11 2023

Formula

a(n) = A007306(A284459(n)), n > 0. - Yosu Yurramendi, Aug 23 2021

A284460 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A245327/A245328 into the enumeration system A002487/A002487' (Calkin-Wilf), and A020651/A020650 (Yu-Ting inverted) into A162911/A162912(Drib).

Original entry on oeis.org

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

Offset: 1

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Author

Yosu Yurramendi, Mar 28 2017

Keywords

nonn

Comments

The inverse permutation is A284459.

Links

Yosu Yurramendi, Table of n, a(n) for n = 1..16383

Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A002487, A020650, A020651, A162911, A162912, A245327, A245328, A284459.

Programs

R

maxrow <- 4 # by choice a <- 1 b01 <- 1 for(m in 0:maxrow){ b01 <- c(b01,rep(1,2^(m+1))); b01[(2^(m+1)+2^m-2^(m-1)):(2^(m+1)+2^m+2^(m-1)-1)] <- 0 for(k in 0:(2^m-1)){ a[2^(m+1) + k] <- a[2^m + k] + 2^(m + b01[2^(m+1) + k]) a[2^(m+1) + 2^m + k] <- a[2^m + k] + 2^(m + b01[2^(m+1) + 2^m + k]) }} a # Yosu Yurramendi, Mar 28 2017

Formula

a(n) = A231550(A258996(n)) = A092569(A231550(n)), n > 0 . - Yosu Yurramendi, Apr 10 2017

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

Original entry on oeis.org

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

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Author

Yosu Yurramendi, Feb 23 2020

Keywords

nonn

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.

Examples

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

Links

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

Crossrefs

Similar R-programs: A258996, A284447.

Programs

PARI

a(n) = bitxor(n, 2<Kevin Ryde, Mar 30 2021

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

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).

Showing 1-8 of 8 results.

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cp's OEIS Frontend

A245327 Numerators in recursive bijection from positive integers to positive rationals, where the bijection is f(1) = 1, f(2n) = 1/(f(n)+1), f(2n+1) = f(n)+1.

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A231551 Position of n in A231550.

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A245328 Denominators in recursive bijection from positive integers to positive rationals (the bijection is f(1) = 1, f(2n) = 1/(f(n)+1), f(2n+1) = f(n)+1).

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

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A154437 Permutation of nonnegative integers: A059893-conjugate of A154435.

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A273493 a(n) = A245327(n) + A245328(n).

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A284460 Permutation of the positive integers: this permutation transforms the enumeration system of positive irreducible fractions A245327/A245328 into the enumeration system A002487/A002487' (Calkin-Wilf), and A020651/A020650 (Yu-Ting inverted) into A162911/A162912(Drib).

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