A154436 -id:A154436 - cp's OEIS frontend

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

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

A153141 Permutation of nonnegative integers: A059893-conjugate of A153151.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13, 8, 9, 10, 11, 31, 30, 28, 29, 24, 25, 26, 27, 16, 17, 18, 19, 20, 21, 22, 23, 63, 62, 60, 61, 56, 57, 58, 59, 48, 49, 50, 51, 52, 53, 54, 55, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 127, 126, 124, 125, 120, 121

Offset: 0

Antti Karttunen, Dec 20 2008

This permutation is induced by a wreath recursion a = s(a,b), b = (b,b) (i.e., binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on page 103 of the Bondarenko, Grigorchuk, et al. paper, starting from the active (swapping) state a and rewriting bits from the second most significant bit to the least significant end, continuing complementing as long as the first 1-bit is reached, which is the last bit to be complemented.
The automorphism group of infinite binary tree (isomorphic to an infinitely iterated wreath product of cyclic groups of two elements) embeds naturally into the group of "size-preserving Catalan bijections". Scheme-function psi gives an isomorphism that maps this kind of permutation to the corresponding Catalan automorphism/bijection (that acts on S-expressions). The following identities hold: *A069770 = psi(A063946) (just swap the left and right subtrees of the root), *A057163 = psi(A054429) (reflect the whole tree), *A069767 = psi(A153141), *A069768 = psi(A153142), *A122353 = psi(A006068), *A122354 = psi(A003188), *A122301 = psi(A154435), *A122302 = psi(A154436) and from *A154449 = psi(A154439) up to *A154458 = psi(A154448). See also comments at A153246 and A153830.
a(1) to a(2^n) is the sequence of row sequency numbers in a Hadamard-Walsh matrix of order 2^n, when constructed to give "dyadic" or Payley sequency ordering. - Ross Drewe, Mar 15 2014
In the Stern-Brocot enumeration system for positive rationals (A007305/A047679), this permutation converts the denominator into the numerator: A007305(n) = A047679(a(n)). - Yosu Yurramendi, Aug 01 2020

			18 = 10010 in binary and after complementing the second, third and fourth most significant bits at positions 3, 2 and 1, we get 1110, at which point we stop (because bit-1 was originally 1) and fix the rest, so we get 11100 (28 in binary), thus a(18)=28. This is the inverse of "binary adding machine". See pages 8, 9 and 103 in the Bondarenko, Grigorchuk, et al. paper.
19 = 10011 in binary. By complementing bits in (zero-based) positions 3, 2 and 1 we get 11101 in binary, which is 29 in decimal, thus a(19)=29.

A. Karttunen, Table of n, a(n) for n = 0..2047
Ievgen Bondarenko, Rostislav Grigorchuk, Rostyslav Kravchenko, Yevgen Muntyan, Volodymyr Nekrashevych, Dmytro Savchuk, and Zoran Sunic, Classification of groups generated by 3-state automata over a 2-letter alphabet, arXiv:0803.3555 [math.GR], 2008, pp. 8--9 & 103.
S. Wolfram and R. Lamy, Discussion on the NKS Forum
Index entries for sequences that are permutations of the natural numbers

Inverse: A153142. a(n) = A059893(A153151(A059893(n))) = A059894(A153152(A059894(n))) = A154440(A154445(n)) = A154442(A154443(n)). Corresponds to A069767 in the group of Catalan bijections. Cf. also A154435-A154436, A154439-A154448, A072376.
Differs from A006068 for the first time at n=14, where a(14)=10 while A006068(14)=11.
A240908-A240910 these give "natural" instead of "dyadic" sequency ordering values for Hadamard-Walsh matrices, orders 8,16,32. - Ross Drewe, Mar 15 2014

def ok(n): return n&(n - 1)==0
def a153151(n): return n if n<2 else 2*n - 1 if ok(n) else n - 1
def A(n): return (int(bin(n)[2:][::-1], 2) - 1)/2
def msb(n): return n if n<3 else msb(n/2)*2
def a059893(n): return A(n) + msb(n)
def a(n): return 0 if n==0 else a059893(a153151(a059893(n))) # Indranil Ghosh, Jun 09 2017

maxlevel <- 5 # by choice
a <- 1
for(m in 1:maxlevel){
a[2^m    ] <- 2^(m+1) - 1
a[2^m + 1] <- 2^(m+1) - 2
for (k in 1:(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}
}
a <- c(0,a)
# Yosu Yurramendi, Aug 01 2020

Conjecture: a(n) = f(a(f(a(A053645(n)))) + A053644(n)) for n > 0 where f(n) = A054429(n) for n > 0 with f(0) = 0. - Mikhail Kurkov, Oct 02 2023
From Mikhail Kurkov, Dec 22 2023: (Start)
a(n) < 2^k iff n < 2^k for k >= 0.
Conjectured formulas:
a(2^m + k) = f(2^m + f(k)) for m >= 0, 0 <= k < 2^m with a(0) = 0.
a(n) = f(A153142(f(n))) for n > 0 with a(0) = 0. (End)

A154435 Permutation of nonnegative integers induced by Lamplighter group generating wreath recursion, variant 3: a = s(b,a), b = (a,b), starting from the state a.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 17 2009

This permutation is induced by the third Lamplighter group generating wreath recursion a = s(b,a), b = (a,b) (i.e., binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on page 104 of Bondarenko, Grigorchuk, et al. paper, starting from the active (swapping) state a and rewriting bits from the second most significant bit to the least significant end.

			475 = 111011011 in binary. Starting from the second most significant bit and, as we begin with the swapping state a, we complement the bits up to and including the first zero encountered and so the beginning of the binary expansion is complemented as 1001....., then, as we switch to the inactive state b, the following bits are kept same, again up to and including the first zero encountered, after which the binary expansion is 1001110.., after which we switch again to the active state (state a), which complements the two rightmost 1's and we obtain the final answer 100111000, which is 312's binary representation, thus a(475)=312.

A. Karttunen, Table of n, a(n) for n = 0..2047
I. Bondarenko, R. Grigorchuk, R. Kravchenko, Y. Muntyan, V. Nekrashevych, D. Savchuk, Z. Sunic, Classification of groups generated by 3-state automata over a 2-letter alphabet, pp. 8--9 & 103, arXiv:0803.3555 [math.GR], 2008.
R. I. Grigorchuk and A. Zuk, The lamplighter group as a group generated by a 2-state automaton and its spectrum, Geometriae Dedicata, vol. 87 (2001), no. 1-3, pp. 209-244.
S. Wolfram, R. Lamy, Discussion on the NKS Forum
Index entries for sequences related to groups
Index entries for sequences that are permutations of the natural numbers

Inverse: A154436. a(n) = A059893(A154437(A059893(n))) = A054429(A006068(A054429(n))). Corresponds to A122301 in the group of Catalan bijections. Cf. also A153141-A153142, A154439-A154448, A072376.

from sympy import floor
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a054429(n): return 1 if n==1 else 2*a054429(floor(n/2)) + 1 - n%2
def a(n): return 0 if n==0 else a054429(a006068(a054429(n))) # Indranil Ghosh, Jun 11 2017

maxn <- 63 # by choice
a <- c(1,3,2) # If it were a <- 1:3, it would be A180200
for(n in 2:maxn){
  a[2*n  ] <- 2*a[n] + (a[n]%%2 == 0)
  a[2*n+1] <- 2*a[n] + (a[n]%%2 != 0)  }
a
# Yosu Yurramendi, Jun 21 2020

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

A154439 Permutation of nonnegative integers induced by Basilica group generating wreath recursion: a = (1,b), b = s(1,a), starting from the inactive (fixing) state a.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 14, 15, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 28, 29, 30, 31, 24, 25, 27, 26, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63, 48, 49, 50, 51, 54, 55, 52, 53, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 0

Antti Karttunen, Jan 17 2009

This permutation is induced by the Basilica group generating wreath recursion a = (1,b), b = s(1,a) (i.e. binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on the page 40 of Bartholdi and Virag paper, starting from the inactive (fixing) state a and rewriting bits from the second most significant bit to the least significant end.

			Starting from the second most significant bit, we continue complementing every second bit (in this case, not starting before at the thirdmost significant bit), as long as the first zero is encountered, which is also complemented if its distance to the most significant bit is even, after which the remaining bits are left intact. E.g. 121 = 1111001 in binary. Complementing its thirdmost significant bit and the first zero-bit two positions right of it (i.e. bit-2, 4 steps to the most significant bit, bit-6), we obtain "11011.." after which the rest of the bits stay same, so we get 1101101, which is 109's binary representation, thus a(121)=109. On the other hand, 125 = 1111101 in binary and the transducer complements the bits at positions 4 and 2, yielding 11010.. and then switches to the fixing state at the zero encounted at bit-position 1, without complementing it (as it is 5 steps from the msb) and the rest are fixed, so we get 1101001, which is 105's binary representation, thus a(125)=105.

R. I. Grigorchuk and A. Zuk, Spectral properties of a torsion free weakly branch group defined by a three state automaton, Contemporary Mathematics 298 (2002), 57--82.

A. Karttunen, Table of n, a(n) for n = 0..2047
L. Bartholdi and B. Virag, Amenability via random walks, arXiv:math/0305262 [math.GR], 2003.
L. Bartholdi and B. Virag, Amenability via random walks, Duke Math. J. Volume 130, Number 1 (2005), 39--56.
Index entries for sequences related to groups
Index entries for sequences that are permutations of the natural numbers

Inverse: A154440. a(n) = A154445(A153142(n)) = A054429(A154443(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154441-A154448. Corresponds to A154449 in the group of Catalan bijections.

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

A154448 Permutation of nonnegative integers induced by wreath recursion a=s(b,c), b=s(c,a), c=(c,c), starting from state a, rewriting bits from the second most significant bit toward the least significant end.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 5, 14, 15, 13, 12, 8, 9, 10, 11, 28, 29, 30, 31, 27, 26, 24, 25, 16, 17, 18, 19, 20, 21, 22, 23, 56, 57, 58, 59, 60, 61, 62, 63, 54, 55, 53, 52, 48, 49, 50, 51, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 112, 113, 114, 115, 116, 117

Offset: 0

Antti Karttunen, Jan 17 2009

This permutation of natural numbers is induced by the first 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 even distance from the most significant bit or B) the first 1-bit at odd distance from the most significant bit.

			25 = 11001 in binary, the first zero-bit at odd distance from the msb is immediately at where we start (at the second most significant bit), so we complement it and fix the rest, yielding 10001 (17 in binary), thus a(25)=17.

Antti Karttunen, Table of n, a(n) for n = 0..2047
Bondarenko, Grigorchuk, Kravchenko, Muntyan, Nekrashevych, Savchuk, and 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: A154447. a(n) = A054429(A154447(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154439-A154446. Corresponds to A154458 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+2)/3)
  a[2*x    ] <- 2*a[x] + 1
  a[2*x + 1] <- 2*a[x]
}
(a <- c(0, a))
# Yosu Yurramendi, Oct 12 2020

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

A233280 Permutation of nonnegative integers: a(n) = A003188(A054429(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 11, 10, 14, 15, 13, 12, 16, 17, 19, 18, 22, 23, 21, 20, 28, 29, 31, 30, 26, 27, 25, 24, 32, 33, 35, 34, 38, 39, 37, 36, 44, 45, 47, 46, 42, 43, 41, 40, 56, 57, 59, 58, 62, 63, 61, 60, 52, 53, 55, 54, 50, 51, 49, 48, 64, 65, 67, 66

Offset: 0

Antti Karttunen, Dec 18 2013

nonn

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

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A233279.
Cf. A000069, A001969, A003188, A054429, A063946, A154436.
Similarly constructed permutation pairs: A003188/A006068, A135141/A227413, A232751/A232752, A233275/A233276, A233277/A233278, A193231 (self-inverse).

from sympy import floor
def a003188(n): return n^(n>>1)
def a054429(n): return 1 if n==1 else 2*a054429(floor(n/2)) + 1 - n%2
def a(n): return 0 if n==0 else a003188(a054429(n)) # Indranil Ghosh, Jun 11 2017

maxrow <- 8 # by choice
a <- 1
for(m in 0:maxrow) for(k in 0:(2^m-1)){
a[2^(m+1)+    k] <- a[2^m+      k] + 2^m
a[2^(m+1)+2^m+k] <- a[2^(m+1)-1-k] + 2^(m+1)
}
a
# Yosu Yurramendi, Apr 05 2017

(define (A233280 n) (A003188 (A054429 n)))
;; Alternative version, based on entangling even & odd numbers with odious and evil numbers:
(definec (A233280 n) (cond ((< n 2) n) ((even? n) (A000069 (+ 1 (A233280 (/ n 2))))) (else (A001969 (+ 1 (A233280 (/ (- n 1) 2)))))))

a(n) = A003188(A054429(n)).
a(n) = A063946(A003188(n)).
a(n) = A054429(A154436(n)).
a(0)=0, a(1)=1, and otherwise, a(2n) = A000069(1+a(n)), a(2n+1) = A001969(1+a(n)). [A recurrence based on entangling even & odd numbers with odious and evil numbers]
a(n) = A258746(A180201(n)) = A180201(A117120(n)), n > 0. - Yosu Yurramendi, Apr 10 2017

A154442 Permutation of nonnegative integers: the inverse of A154441.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 17 2009

Inverse: A154441. a(n) = A153141(A154444(n)) = A054429(A154446(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154439-A154448. Corresponds to A154452 in the group of Catalan bijections.

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

A154446 Permutation of nonnegative integers: The inverse of A154445.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 17 2009

Inverse: A154445. a(n) = A153142(A154440(n)) = A054429(A154442(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154439-A154448. Corresponds to A154456 in the group of Catalan bijections.

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

A154440 Permutation of nonnegative integers: the inverse of A154439.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 14, 15, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 28, 29, 31, 30, 24, 25, 26, 27, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 56, 57, 58, 59, 62, 63, 60, 61, 48, 49, 50, 51, 52, 53, 54, 55, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 0

Antti Karttunen, Jan 17 2009

Inverse: A154439. a(n) = A153141(A154446(n)) = A054429(A154444(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154442-A154448. Corresponds to A154450 in the group of Catalan bijections.

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

A154441 Permutation of nonnegative integers induced by Basilica group generating wreath recursion: a = (1,b), b = s(1,a), starting from the active (swapping) state b.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 17 2009

This permutation is induced by the Basilica group generating wreath recursion a = (1,b), b = s(1,a) (i.e. binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on the page 40 of Bartholdi and Virag paper, starting from the active (switching) state b and rewriting bits from the second most significant bit to the least significant end.

			Starting from the second most significant bit, we continue complementing every second bit (in this case, starting from the second most significant bit), as long as the first zero is encountered, which is also complemented if its distance to the most significant bit is odd, after which the remaining bits are left intact. E.g. 121 = 1111001 in binary. Complementing its second and fourth most significant bits (positions 5 & 3) and stopping at the first zero-bit at position 2 (which is not complemented, as its distance to the msb is 6), we obtain "10100.." after which the rest of the bits stay same, so we get 1010001, which is 81's binary representation, thus a(121)=81. On the other hand, 125 = 1111101 in binary and the transducer complements the bits at positions 5, 3 and also the first zero at the position 1 (because at odd distance from the msb), yielding 101011., after which the remaining bit stays same, thus we get 1010111, which is 87's binary representation, thus a(125)=87.

R. I. Grigorchuk and A. Zuk, Spectral properties of a torsion free weakly branch group defined by a three state automaton, Contemporary Mathematics 298 (2002), 57--82.

A. Karttunen, Table of n, a(n) for n = 0..2047
L. Bartholdi and B. Virag, Amenability via random walks, arXiv:math/0305262 [math.GR], 2003.
L. Bartholdi and B. Virag, Amenability via random walks, Duke Math. J. Volume 130, Number 1 (2005), 39--56.
Index entries for sequences that are permutations of the natural numbers

Inverse: A154442. a(n) = A154443(A153142(n)) = A054429(A154445(A054429(n))). Cf. A072376, A153141-A153142, A154435-A154436, A154439-A154448. Corresponds to A154451 in the group of Catalan bijections.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

R

Formula

A153141 Permutation of nonnegative integers: A059893-conjugate of A153151.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

R

Formula

A154435 Permutation of nonnegative integers induced by Lamplighter group generating wreath recursion, variant 3: a = s(b,a), b = (a,b), starting from the state a.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

R

Extensions

A154439 Permutation of nonnegative integers induced by Basilica group generating wreath recursion: a = (1,b), b = s(1,a), starting from the inactive (fixing) state a.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A154448 Permutation of nonnegative integers induced by wreath recursion a=s(b,c), b=s(c,a), c=(c,c), starting from state a, rewriting bits from the second most significant bit toward the least significant end.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

R

Extensions

A233280 Permutation of nonnegative integers: a(n) = A003188(A054429(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

R

Scheme

Formula

A154442 Permutation of nonnegative integers: the inverse of A154441.

Views

Author

Keywords

Links

Crossrefs

Extensions

A154446 Permutation of nonnegative integers: The inverse of A154445.

Views

Author

Keywords

Links