A154439 -id:A154439 - cp's OEIS frontend

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.

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

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

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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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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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