A117966 -id:A117966 - cp's OEIS frontend

A117967 Positive part of inverse of A117966; write n in balanced ternary and then replace (-1)'s with 2's.

0, 1, 5, 3, 4, 17, 15, 16, 11, 9, 10, 14, 12, 13, 53, 51, 52, 47, 45, 46, 50, 48, 49, 35, 33, 34, 29, 27, 28, 32, 30, 31, 44, 42, 43, 38, 36, 37, 41, 39, 40, 161, 159, 160, 155, 153, 154, 158, 156, 157, 143, 141, 142, 137, 135, 136, 140, 138, 139, 152, 150, 151, 146

Offset: 0

Franklin T. Adams-Watters, Apr 05 2006

			7 in balanced ternary is 1(-1)1, changing to 121 ternary is 16, so a(7)=16.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175

Alois P. Heinz, Table of n, a(n) for n = 0..9841
Ken Levasseur, The Balanced Ternary Number System

Cf. A117966. a(n) = A004488(A117968(n)). Bisection of A140263. A140267 gives the same sequence in ternary.

a:= proc(n) local d, i, m, r; m:=n; r:=0;
      for i from 0 while m>0 do
         d:= irem(m, 3, 'm');
         if d=2 then m:=m+1 fi;
         r:= r+d*3^i
      od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 11 2015

a[n_] := Module[{d, i, m = n, r = 0}, For[i = 0, m > 0, i++, {m, d} = QuotientRemainder[m, 3]; If[d == 2, m++]; r = r + d*3^i]; r];
a /@ Range[0, 100] (* Jean-François Alcover, Jan 05 2021, after Alois P. Heinz *)

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 a(n): return 0 if n==0 else a004488(a117968(n)) # Indranil Ghosh, Jun 06 2017

;; Two alternative definitions in MIT/GNU Scheme, defined for whole Z:
(define (A117967 z) (cond ((zero? z) 0) ((negative? z) (A004488 (A117967 (- z)))) (else (let* ((lp3 (expt 3 (A062153 z))) (np3 (* 3 lp3))) (if (< (* 2 z) np3) (+ lp3 (A117967 (- z lp3))) (+ np3 (A117967 (- z np3))))))))
(define (A117967v2 z) (cond ((zero? z) 0) ((negative? z) (A004488 (A117967v2 (- z)))) ((zero? (modulo z 3)) (* 3 (A117967v2 (/ z 3)))) ((= 1 (modulo z 3)) (+ (* 3 (A117967v2 (/ (- z 1) 3))) 1)) (else (+ (* 3 (A117967v2 (/ (+ z 1) 3))) 2))))
;; Antti Karttunen, May 19 2008

a(0) = 0, a(3n) = 3a(n), a(3n+1) = 3a(n)+1, a(3n-1) = 3a(n)+2.

If one adds this clause, then the function is defined on the whole Z: If n<0, then a(n) = A004488(a(-n)) (or equivalently: a(n) = A117968(-n)) and then it holds that a(A117966(n)) = n. - Antti Karttunen, May 19 2008

A117968 Negative part of inverse of A117966; write -n in balanced ternary and then replace (-1)'s with 2's.

Original entry on oeis.org

2, 7, 6, 8, 22, 21, 23, 19, 18, 20, 25, 24, 26, 67, 66, 68, 64, 63, 65, 70, 69, 71, 58, 57, 59, 55, 54, 56, 61, 60, 62, 76, 75, 77, 73, 72, 74, 79, 78, 80, 202, 201, 203, 199, 198, 200, 205, 204, 206, 193, 192, 194, 190, 189, 191, 196, 195, 197, 211, 210, 212, 208, 207

Offset: 1

Franklin T. Adams-Watters, Apr 05 2006

			-7 in balanced ternary is (-1)1(-1), changing to 212 ternary is 23, so a(7)=23.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175

Indranil Ghosh, Table of n, a(n) for n = 1..6561
Ken Levasseur, The Balanced Ternary Number System

Cf. A117966. a(n) = A004488(A117967(n)). Bisection of A140263. A140268 gives the same sequence in ternary.

def a(n):
    if n==1: return 2
    if n%3==0: return 3*a(n//3)
    elif n%3==1: return 3*a((n - 1)//3) + 2
    else: return 3*a((n + 1)//3) + 1
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

a(1) = 2, a(3n) = 3a(n), a(3n+1) = 3a(n)+2, a(3n-1) = 3a(n)+1.

A365711 Dirichlet inverse of balanced ternary enumeration of integers (A117966).

Original entry on oeis.org

1, 1, -3, -3, -2, -3, 2, -3, 0, -14, -8, 9, -13, -7, 6, 6, -5, 0, 8, 0, -6, -11, 7, 9, 15, -13, 0, -60, -26, 42, -31, -30, 24, -35, -31, 0, -37, -19, 39, 54, -38, 21, -34, 18, 0, -5, -17, -18, -18, 54, 15, 75, -14, 0, 58, 3, -24, -29, 25, 0, 29, -31, 0, -57, 71, 33, 14, -9, -21, -46, 22, 0, 35, -37, -45, -78, 2, 39

Offset: 1

Antti Karttunen, Sep 19 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A011655, A117966, A359377, A365428, A365712.

Cf. also A365713, A365803.

A117966(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n, 3)), 'x), 'x, 3); \\ From A117966 by Gheorghe Coserea
memoA365711 = Map();
A365711(n) = if(1==n,1,my(v); if(mapisdefined(memoA365711,n,&v), v, v = -sumdiv(n,d,if(dA117966(n/d)*A365711(d),0)); mapput(memoA365711,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA117966(n/d) * a(d).

A011655(abs(a(n))) = A359377(n).

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).

Original entry on oeis.org

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)).

A246210 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 1 + 2a(-(A117966(n))), otherwise a(n) = 2a(A117966(n)-1).

Original entry on oeis.org

0, 1, 3, 6, 12, 2, 13, 7, 25, 50, 100, 14, 28, 56, 200, 4, 26, 24, 101, 51, 201, 27, 5, 15, 57, 29, 113, 226, 452, 58, 116, 232, 904, 30, 114, 10, 20, 40, 228, 456, 912, 80, 1808, 60, 464, 48, 202, 52, 402, 54, 102, 400, 8, 112, 453, 227, 905, 115, 31, 59, 233, 117, 465, 203, 49, 103, 9, 401, 53, 55, 403, 11, 41, 21, 81, 61

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 complementary pair A005843/A005408 (even and odd numbers respectively), with a(0) set to 0 and a(1) set to 1.

This implies that apart from a(1) = 1, even numbers occur only in positions given by A117967, and odd numbers only in positions given by A117968.

Antti Karttunen, Table of n, a(n) for n = 0..2187
Index entries for sequences that are permutations of the natural numbers

Inverse: A246209.
Similar or related permutations: A054429, A246208, 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 1 + 2*a(-x)
    else: return 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) = 1 + 2*a(-(A117966(n))), otherwise a(n) = 2*a(A117966(n)-1).
As a composition of related permutations:
a(n) = A054429(A246208(n)).
a(n) = A246208(A246211(n)).

A338245 Nonnegative values in A117966, in order of appearance.

Original entry on oeis.org

0, 1, 3, 4, 2, 9, 10, 8, 12, 13, 11, 6, 7, 5, 27, 28, 26, 30, 31, 29, 24, 25, 23, 36, 37, 35, 39, 40, 38, 33, 34, 32, 18, 19, 17, 21, 22, 20, 15, 16, 14, 81, 82, 80, 84, 85, 83, 78, 79, 77, 90, 91, 89, 93, 94, 92, 87, 88, 86, 72, 73, 71, 75, 76, 74, 69, 70, 68

Offset: 0

Rémy Sigrist, Oct 18 2020

This sequence is a permutation of the nonnegative integers with inverse A338247 (the offset has been set to 0 so as to get a permutation).

There are only two fixed points: a(0) = 0 and a(1) = 1.

			A117966 = 0, 1, -1, 3, 4, 2, -3, -2, -4, 9, 10, 8, 12, 13, 11, 6, 7, 5, ...
We keep:  0, 1,     3, 4, 2,             9, 10, 8, 12, 13, 11, 6, 7, 5, ...

See A338248 for a similar sequence.

Cf. A117966, A132141, A338246, A338247.

A117966(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n, 3)), 'x), 'x, 3)
print (select(v -> v>=0, apply(A117966, [0..107])))

a(0) = 0.

a(n) = A117966(A132141(n)) for any n > 0.

A338246 Nonpositive values in A117966, in order of appearance and negated.

Original entry on oeis.org

0, 1, 3, 2, 4, 9, 8, 10, 6, 5, 7, 12, 11, 13, 27, 26, 28, 24, 23, 25, 30, 29, 31, 18, 17, 19, 15, 14, 16, 21, 20, 22, 36, 35, 37, 33, 32, 34, 39, 38, 40, 81, 80, 82, 78, 77, 79, 84, 83, 85, 72, 71, 73, 69, 68, 70, 75, 74, 76, 90, 89, 91, 87, 86, 88, 93, 92, 94

Offset: 0

Rémy Sigrist, Oct 18 2020

This sequence is a self-inverse permutation of the nonnegative integers (the offset has been set to 0 so as to get a permutation).

			A117966 = 0, 1, -1, 3, 4, 2, -3, -2, -4, 9, 10, 8, 12, 13, 11, 6, 7, 5, -9, ...
We keep:  0,     1,           3,  2,  4,                                 9, ...

Cf. A003462 (fixed points), A117966, A157671, A338245.

A117966(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n, 3)), 'x), 'x, 3)
print (-select(v -> v<=0, apply(A117966, [0..188])))

a(0) = 0.
a(n) = -A117966(A157671(n)) for any n > 0.
a(n) = n iff n belongs to A003462.

A365712 Sum of balanced ternary enumeration of integers (A117966) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 1, 0, -6, 0, -7, 9, -4, 0, 21, 0, 4, 12, 13, 0, -9, 0, -10, -12, -16, 0, -3, 4, -26, 27, -32, 0, 72, 0, -1, 48, -10, -8, 36, 0, 16, 78, 94, 0, 54, 0, 50, 18, 14, 0, 3, 4, 74, 30, 91, 0, -27, 32, -25, -48, -52, 0, -30, 0, -62, -18, -74, 52, 18, 0, -25, -42, -66, 0, -36, 0, -74, -78, -110, -32, 0, 0, -40, 81

Offset: 1

Antti Karttunen, Sep 19 2023

sign

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A117966, A365711.

Cf. also A365714, A365804.

A117966(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n, 3)), 'x), 'x, 3); \\ From A117966 by Gheorghe Coserea
memoA365711 = Map();
A365711(n) = if(1==n,1,my(v); if(mapisdefined(memoA365711,n,&v), v, v = -sumdiv(n,d,if(dA117966(n/d)*A365711(d),0)); mapput(memoA365711,n,v); (v)));
A365712(n) = (A117966(n)+A365711(n));

a(n) = A117966(n) + A365711(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A117966(d) * A365711(n/d).

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

A117967 Positive part of inverse of A117966; write n in balanced ternary and then replace (-1)'s with 2's.

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A117968 Negative part of inverse of A117966; write -n in balanced ternary and then replace (-1)'s with 2's.

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A365711 Dirichlet inverse of balanced ternary enumeration of integers (A117966).

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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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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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A246210 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 1 + 2*a(-(A117966(n))), otherwise a(n) = 2*a(A117966(n)-1).

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A338245 Nonnegative values in A117966, in order of appearance.

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A338246 Nonpositive values in A117966, in order of appearance and negated.

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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).

A246210 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, and for n > 1, if A117966(n) < 1, a(n) = 1 + 2a(-(A117966(n))), otherwise a(n) = 2a(A117966(n)-1).