A246210 -id:A246210 - cp's OEIS frontend

A246208 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 2a(-(A117966(n))), otherwise a(n) = 1 + 2a(A117966(n)-1).

0, 1, 2, 5, 11, 3, 10, 4, 22, 45, 91, 9, 19, 39, 183, 7, 21, 23, 90, 44, 182, 20, 6, 8, 38, 18, 78, 157, 315, 37, 75, 151, 631, 17, 77, 13, 27, 55, 155, 311, 623, 111, 1263, 35, 303, 47, 181, 43, 365, 41, 89, 367, 15, 79, 314, 156, 630, 76, 16, 36, 150, 74, 302, 180, 46, 88, 14, 366, 42, 40, 364, 12, 54, 26, 110, 34

Offset: 0

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where complementary pair A117968/A117967 (negative and positive part of inverse of balanced ternary enumeration of integers, respectively) is entangled with complementary pair A005843/A005408 (even and odd numbers respectively), with a(0) set to 0 and a(1) set to 1.

Thus this shares with A140264 the property that apart from a(0) = 0, even numbers occur only in positions given by A117968, and odd numbers only in positions given by A117967.

Antti Karttunen, Table of n, a(n) for n = 0..6561
Index entries for sequences that are permutations of the natural numbers

Inverse: A246207.
Related permutations: A140264, A054429, A246210, A246211.
Cf. A005408, A005843, A117966, A117967, A117968.

def a117966(n):
    if n==0: return 0
    if n%3==0: return 3*a117966(n//3)
    elif n%3==1: return 3*a117966((n - 1)//3) + 1
    else: return 3*a117966((n - 2)//3) - 1
def a(n):
    if n<2: return n
    x=a117966(n)
    if x<1: return 2*a(-x)
    else: return 1 + 2*a(x - 1)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 2*a(-(A117966(n))), otherwise a(n) = 1 + 2*a(A117966(n)-1).
As a composition of related permutations:
a(n) = A054429(A246210(n)).
a(n) = A246210(A246211(n)).

A246211 Self-inverse permutation of natural numbers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 0, a(n) = A117967(1+a(-(A117966(n)))), otherwise a(n) = A117968(a(A117966(n)-1)).

Original entry on oeis.org

0, 1, 5, 22, 71, 2, 35, 15, 99, 225, 531, 66, 213, 516, 1899, 7, 73, 172, 307, 127, 1369, 36, 3, 52, 304, 148, 1246, 5408, 17461, 620, 1567, 5321, 41591, 194, 698, 6, 21, 69, 1489, 5165, 16975, 174, 142234, 643, 17287, 587, 695, 173, 5195, 72, 605, 4770, 23, 1761, 12051, 4175, 24134, 389, 137, 431, 3758, 945, 11964, 392, 419, 482, 11, 2872, 104, 37, 3830, 4, 49, 16

Offset: 0

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where complementary pair A117967/A117968 (positive and negative part of inverse of balanced ternary enumeration of integers, respectively) is entangled with the same pair in the opposite order: A117967/A117968, with a(0) set to 0 and a(1) set to 1.

Antti Karttunen, Table of n, a(n) for n = 0..2187
Indranil Ghosh, Python program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Related or similar permutations: A246207, A246208, A246209, A246210, A004488, A245812, A054429.

Cf. A117966, A117967, A117968.

a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 0, a(n) = A117967(1+a(-(A117966(n)))), otherwise a(n) = A117968(a(A117966(n)-1)).

A246209 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, a(2n) = A117967(1+a(n)), a(2n+1) = A117968(a(n)).

Original entry on oeis.org

0, 1, 5, 2, 15, 22, 3, 7, 52, 66, 35, 71, 4, 6, 11, 23, 137, 194, 148, 213, 36, 73, 99, 172, 17, 8, 16, 21, 12, 25, 33, 58, 462, 601, 447, 643, 431, 620, 304, 516, 37, 72, 104, 173, 127, 225, 419, 587, 45, 64, 9, 19, 47, 68, 49, 69, 13, 24, 29, 59, 43, 75, 152, 197, 1273, 1734, 1334, 1940, 1294, 1740, 899, 1556, 1404, 1837, 945, 1567, 389, 698, 1246, 1761, 41

Offset: 0

Antti Karttunen, Aug 19 2014

nonn

This is an instance of entanglement permutation, where complementary pair A005843/A005408 (even and odd numbers respectively) is entangled with complementary pair A117967/A117968 (positive and negative part of inverse of balanced ternary enumeration of integers, respectively), with a(0) set to 0 and a(1) set to 1.

This implies that the even positions contain only terms of A117967 and apart from a(1) = 1, the odd positions contain only terms of A117968.

Antti Karttunen, Table of n, a(n) for n = 0..2047
Index entries for sequences that are permutations of the natural numbers

Inverse: A246210.
Related permutations: A054429, A246207, A246211.
Cf. A117967, A117968.

from sympy.ntheory.factor_ import digits
def a004488(n): return int("".join(str((3 - i)%3) for i in digits(n, 3)[1:]), 3)
def a117968(n):
    if n==1: return 2
    if n%3==0: return 3*a117968(n//3)
    elif n%3==1: return 3*a117968((n - 1)//3) + 2
    else: return 3*a117968((n + 1)//3) + 1
def a117967(n): return 0 if n==0 else a117968(-n) if n<0 else a004488(a117968(n))
def a(n): return n if n<2 else a117967(1 + a(n//2)) if n%2==0 else a117968(a((n - 1)//2))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

a(0) = 0, a(1) = 1, a(2n) = A117967(1+a(n)), a(2n+1) = A117968(a(n)).
As a composition of related permutations:
a(n) = A246207(A054429(n)).
a(n) = A246211(A246207(n)).

A246206 Permutation of natural numbers: a(1) = 1, if A117966(n) < 0, a(n) = A014580(a(-(A117966(n)))), otherwise a(n) = A091242(a(A117966(n)-1)).

Original entry on oeis.org

1, 2, 5, 9, 4, 13, 3, 37, 49, 64, 6, 10, 16, 81, 8, 20, 15, 351, 229, 451, 59, 11, 7, 41, 19, 73, 92, 114, 27, 36, 48, 140, 12, 53, 17, 24, 33, 69, 86, 107, 44, 170, 18, 63, 22, 410, 28, 524, 76, 271, 101, 14, 23, 687, 529, 895, 253, 25, 97, 213, 145, 333, 3413, 67, 2091, 31, 607, 103, 415, 4531, 47, 131, 87, 193, 55

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

Compare to the formula for A246164. However, instead of reversing binary representation, we employ here balanced ternary enumeration of integers (see A117966).

Inverse: A246205.
Similar or related entanglement permutations: A246164, A245702, A246202, A246208, A246210.
Cf. A014580, A091242, A117966.

a(1) = 1, and for n > 1, if A117966(n) < 0, then a(n) = A014580(a(-(A117966(n)))), otherwise a(n) = A091242(a(A117966(n)-1)).
As a composition of related permutations:
a(n) = A245702(A246208(n)).
a(n) = A246202(A246210(n)).

cp's OEIS Frontend

A246208 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 2a(-(A117966(n))), otherwise a(n) = 1 + 2a(A117966(n)-1).

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A246211 Self-inverse permutation of natural numbers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 0, a(n) = A117967(1+a(-(A117966(n)))), otherwise a(n) = A117968(a(A117966(n)-1)).

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A246209 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, a(2n) = A117967(1+a(n)), a(2n+1) = A117968(a(n)).

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Python

Formula

A246206 Permutation of natural numbers: a(1) = 1, if A117966(n) < 0, a(n) = A014580(a(-(A117966(n)))), otherwise a(n) = A091242(a(A117966(n)-1)).

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cp's OEIS Frontend

A246208 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 2*a(-(A117966(n))), otherwise a(n) = 1 + 2*a(A117966(n)-1).

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Author

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Crossrefs

Programs

Python

Formula

A246211 Self-inverse permutation of natural numbers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 0, a(n) = A117967(1+a(-(A117966(n)))), otherwise a(n) = A117968(a(A117966(n)-1)).

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Formula

A246209 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, a(2n) = A117967(1+a(n)), a(2n+1) = A117968(a(n)).

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Author

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Programs

Python

Formula

A246206 Permutation of natural numbers: a(1) = 1, if A117966(n) < 0, a(n) = A014580(a(-(A117966(n)))), otherwise a(n) = A091242(a(A117966(n)-1)).

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Comments

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Formula

A246208 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 2a(-(A117966(n))), otherwise a(n) = 1 + 2a(A117966(n)-1).