A154436 -id:A154436 - cp's OEIS frontend

A154443 Permutation of nonnegative integers induced by Basilica group generating wreath recursion: a = (b,1), b = s(a,1), starting from the inactive (fixing) state a.

0, 1, 2, 3, 5, 4, 6, 7, 10, 11, 8, 9, 12, 13, 14, 15, 21, 20, 22, 23, 16, 17, 18, 19, 24, 25, 26, 27, 28, 29, 30, 31, 42, 43, 40, 41, 44, 45, 46, 47, 32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 85, 84, 86, 87, 80, 81, 82, 83

Offset: 0

Antti Karttunen, Jan 17 2009

Inverse: A154444. a(n) = A154441(A153141(n)) = A054429(A154439(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154439-A154448. Corresponds to A154453 in the group of Catalan bijections.

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A154445 Permutation of nonnegative integers induced by Basilica group generating wreath recursion: a = (b,1), b = s(a,1), starting from the active (swapping) state b.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 4, 5, 13, 12, 14, 15, 8, 9, 10, 11, 26, 27, 24, 25, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 53, 52, 54, 55, 48, 49, 50, 51, 56, 57, 58, 59, 60, 61, 62, 63, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 106, 107, 104, 105, 108, 109

Offset: 0

Antti Karttunen, Jan 17 2009

Inverse: A154446. a(n) = A154439(A153141(n)) = A054429(A154441(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154439-A154448. Corresponds to A154455 in the group of Catalan bijections.

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A180201 Inverse permutation to A180200.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 11, 10, 8, 9, 13, 12, 14, 15, 23, 22, 20, 21, 17, 16, 18, 19, 27, 26, 24, 25, 29, 28, 30, 31, 47, 46, 44, 45, 41, 40, 42, 43, 35, 34, 32, 33, 37, 36, 38, 39, 55, 54, 52, 53, 49, 48, 50, 51, 59, 58, 56, 57, 61, 60, 62, 63, 95, 94, 92, 93, 89, 88, 90, 91, 83

Offset: 0

Reinhard Zumkeller, Aug 15 2010

A180199(n) = a(a(n));
a(A180198(n)) = A180198(a(n)) = A180200(n);
a(A075427(n)) = A075427(n).
This permutation transforms the enumeration system of positive irreducible fractions A245325/A245326 into the enumeration system A007305/A047679 (Stern-Brocot), and enumeration system A071766/A229742 (HCS) into A162909/A162910 (Bird). - Yosu Yurramendi, Jun 09 2015

Index entries for sequences that are permutations of the natural numbers

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

a(n) = A233280(A258746(n)) = A117120(A233280(n)), n > 0. - Yosu Yurramendi, Apr 10 2017 [Corrected by Yosu Yurramendi, Mar 14 2025]
a(0) = 0, a(1) = 1, for n > 0 a(2*n) = 2*a(n) + [n even], a(2*n + 1) = 2*a(n) + [n odd]. - Yosu Yurramendi, May 23 2020
From Alan Michael Gómez Calderón, Mar 04 2025: (Start)
a(n) = A054429(n) XOR floor(n/2) for n > 0.
a(n) = A054429(A003188(n)) for n > 0. (End)
a(n) = A154436(A054429(n)), n > 0. - Yosu Yurramendi, Mar 11 2025

A154444 Permutation of nonnegative integers: The inverse of A154443.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 10, 11, 8, 9, 12, 13, 14, 15, 20, 21, 22, 23, 17, 16, 18, 19, 24, 25, 26, 27, 28, 29, 30, 31, 40, 41, 42, 43, 44, 45, 46, 47, 34, 35, 32, 33, 36, 37, 38, 39, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 0

Antti Karttunen, Jan 17 2009

Inverse: A154443. a(n) = A153142(A154442(n)) = A054429(A154440(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154439-A154448. Corresponds to A154454 in the group of Catalan bijections.

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A154447 Permutation of nonnegative integers induced by wreath recursion a=s(b,c), b=s(c,a), c=(c,c), starting from state b, rewriting bits from the second most significant bit toward the least significant end.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 14, 15, 11, 10, 8, 9, 24, 25, 26, 27, 28, 29, 30, 31, 22, 23, 21, 20, 16, 17, 18, 19, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 44, 45, 46, 47, 43, 42, 40, 41, 32, 33, 34, 35, 36, 37, 38, 39, 96, 97, 98, 99, 100, 101, 102

Offset: 0

Antti Karttunen, Jan 17 2009

This permutation of natural numbers is induced by the second generator of group 2861 mentioned on page 144 of "Classification of groups generated by 3-state automata over a 2-letter alphabet" paper. It can be computed by starting scanning n's binary expansion rightward from the second most significant bit, complementing every bit down to and including A) either the first 0-bit at odd distance from the most significant bit or B) the first 1-bit at even distance from the most significant bit.

			25 = 11001 in binary, the first zero-bit at odd distance from the msb is at position 1 (distance 3) and the first one-bit at even distance from the msb is at position 0 (distance 4), thus we stop at the former, after complementing the bits 3-1, which gives us 10111 (23 in binary), thus a(25)=23.

A. Karttunen, Table of n, a(n) for n = 0..2047
Bondarenko, Grigorchuk, Kravchenko, Muntyan, Nekrashevych, Savchuk, Sunic, Classification of groups generated by 3-state automata over a 2-letter alphabet, arXiv:0803.3555 [math.GR], 2008, p. 144.
Index entries for sequences that are permutations of the natural numbers

Inverse: A154448. a(n) = A054429(A154448(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154439-A154446. Corresponds to A154457 in the group of Catalan bijections.

maxlevel <- 5 # by choice
a <- 1
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+1) + 2*k + 1] <- 2*a[2^m + k] + 1
  }
  x <- floor(2^m*5/3)
  a[2*x    ] <- 2*a[x] + 1
  a[2*x + 1] <- 2*a[x]
}
(a <- c(0, a))
# Yosu Yurramendi, Oct 12 2020

cp's OEIS Frontend

A154443 Permutation of nonnegative integers induced by Basilica group generating wreath recursion: a = (b,1), b = s(a,1), starting from the inactive (fixing) state a.

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A154445 Permutation of nonnegative integers induced by Basilica group generating wreath recursion: a = (b,1), b = s(a,1), starting from the active (swapping) state b.

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A180201 Inverse permutation to A180200.

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A154444 Permutation of nonnegative integers: The inverse of A154443.

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A154447 Permutation of nonnegative integers induced by wreath recursion a=s(b,c), b=s(c,a), c=(c,c), starting from state b, rewriting bits from the second most significant bit toward the least significant end.

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R