A117967 -id:A117967 - cp's OEIS frontend

A140263 Permutation of nonnegative integers obtained by interleaving A117967 and A117968.

0, 1, 2, 5, 7, 3, 6, 4, 8, 17, 22, 15, 21, 16, 23, 11, 19, 9, 18, 10, 20, 14, 25, 12, 24, 13, 26, 53, 67, 51, 66, 52, 68, 47, 64, 45, 63, 46, 65, 50, 70, 48, 69, 49, 71, 35, 58, 33, 57, 34, 59, 29, 55, 27, 54, 28, 56, 32, 61, 30, 60, 31, 62, 44, 76, 42, 75, 43, 77, 38, 73, 36

Offset: 0

Antti Karttunen, May 19 2008, originally described in a posting at the SeqFan mailing list on Sep 15 2005

Inverse: A140264. Bisections: A117967 & A117968. a(n) = A140265(n+1)-1.

Cf. A004488, A117966.

from sympy import ceiling
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 a001057(n): return -(-1)**n*ceiling(n/2)
def a(n): return a117967(a001057(n)) # Indranil Ghosh, Jun 07 2017

a(n) = A117967(A001057(n)). (Assuming that the domain of A117967 is the whole Z line.)

A246207 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, a(2n) = A117968(a(n)), a(2n+1) = A117967(1+a(n)).

Original entry on oeis.org

0, 1, 2, 5, 7, 3, 22, 15, 23, 11, 6, 4, 71, 35, 66, 52, 58, 33, 25, 12, 21, 16, 8, 17, 172, 99, 73, 36, 213, 148, 194, 137, 197, 152, 75, 43, 59, 29, 24, 13, 69, 49, 68, 47, 19, 9, 64, 45, 587, 419, 225, 127, 173, 104, 72, 37, 516, 304, 620, 431, 643, 447, 601, 462, 640, 441, 577, 423, 177, 103, 203, 155, 211, 150, 61, 30, 57, 34, 26, 53

Offset: 0

Antti Karttunen, Aug 19 2014

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

Thus this shares with A140263 the property that after a(0)=0, the even positions contain only terms of A117968 and the odd positions contain only terms of A117967.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Pin & logarithmic scatter plots computed for range 0..2047 only (computed with OEIS "graph" command)
Index entries for sequences that are permutations of the natural numbers

Inverse: A246208.
Related permutations: A140263, A054429, A246209, 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 a117968(a(n/2)) if n%2==0 else a117967(1 + a((n - 1)/2)) # Indranil Ghosh, Jun 07 2017

As a composition of related permutations:
a(n) = A246209(A054429(n)).
a(n) = A246211(A246209(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)).

A246205 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A117968(a(n)), a(A091242(n)) = A117967(1+a(n)), where A117967 and A117968 give positive and negative parts of inverse of balanced ternary enumeration of integers, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

Original entry on oeis.org

1, 2, 7, 5, 3, 11, 23, 15, 4, 12, 22, 33, 6, 52, 17, 13, 35, 43, 25, 16, 137, 45, 53, 36, 58, 155, 29, 47, 462, 154, 66, 135, 37, 152, 426, 30, 8, 156, 1273, 428, 24, 148, 460, 41, 423, 1426, 71, 31, 9, 427, 4283, 1410, 34, 431, 75, 1274, 159, 1423, 21, 3707, 194, 99, 44, 10, 1412, 11115, 64, 3850, 38, 1404, 103, 4281, 26, 412, 3722, 49

Offset: 1

Antti Karttunen, Aug 19 2014

nonn

Inverse: A246206.
Similar or related entanglement permutations: A246163, A245701, A246201, A246207, A246209.
Cf. A014580, A091225, A091226, A091242, A091245, A117967, A117968.

a(1) = 1, and for n > 1, if A091225(n) = 1 [i.e. n is in A014580], a(n) = A117968(a(A091226(n))), otherwise a(n) = A117967(1+a(A091245(n))).
As a composition of related permutations:
a(n) = A246207(A245701(n)).
a(n) = A246209(A246201(n)).

A004488 Tersum n + n.

Original entry on oeis.org

0, 2, 1, 6, 8, 7, 3, 5, 4, 18, 20, 19, 24, 26, 25, 21, 23, 22, 9, 11, 10, 15, 17, 16, 12, 14, 13, 54, 56, 55, 60, 62, 61, 57, 59, 58, 72, 74, 73, 78, 80, 79, 75, 77, 76, 63, 65, 64, 69, 71, 70, 66, 68, 67, 27, 29, 28, 33, 35, 34, 30, 32, 31, 45, 47, 46, 51

Offset: 0

N. J. A. Sloane

Could also be described as "Write n in base 3, then replace each digit with its base-3 negative" as with A048647 for base 4. - Henry Bottomley, Apr 19 2000
a(a(n)) = n, a self-inverse permutation of the nonnegative integers. - Reinhard Zumkeller, Dec 19 2003
First 3^n terms of the sequence form a permutation s(n) of 0..3^n-1, n>=1; the number of inversions of s(n) is A016142(n-1). - Gheorghe Coserea, Apr 23 2018

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 6561 terms from Alois P. Heinz)
Index entries for sequences related to carryless arithmetic
Index entries for sequences that are permutations of the natural numbers

Column k=0 of A253586, A253587.
Column k=3 of A248813.
Row / column 2 of A325820.
Main diagonal of A004489.
Cf. A048647, A055115, A055116, A055120, A059249, A117966, A117967, A117968, A225901, A242399, A244042, A263273, A289813, A289814, A289815, A289816, A289831, A289838, A300222, A321464.

a004488 0 = 0
a004488 n = if d == 0 then 3 * a004488 n' else 3 * a004488 n' + 3 - d
            where (n', d) = divMod n 3
-- Reinhard Zumkeller, Mar 12 2014

a:= proc(n) local t, r, i;
      t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+3^i *irem(2*irem(t, 3, 't'), 3)
      od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 07 2011

a[n_] := FromDigits[Mod[3-IntegerDigits[n, 3], 3], 3]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Mar 03 2014 *)

a(n) = my(b=3); fromdigits(apply(d->(b-d)%b, digits(n, b)), b);
vector(67, i, a(i-1))  \\ Gheorghe Coserea, Apr 23 2018

from sympy.ntheory.factor_ import digits
def a(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3) # Indranil Ghosh, Jun 06 2017

Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g., 5 + 8 = "21" + "22" = "10" = 1.
a(n) = Sum(3-d(i)-3*0^d(i): n=Sum(d(i)*3^d(i): 0<=d(i)<3)). - Reinhard Zumkeller, Dec 19 2003
a(3*n) = 3*a(n), a(3*n+1) = 3*a(n)+2, a(3*n+2) = 3*a(n)+1. - Robert Israel, May 09 2014

A117966 Balanced ternary enumeration (based on balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'s.

Original entry on oeis.org

0, 1, -1, 3, 4, 2, -3, -2, -4, 9, 10, 8, 12, 13, 11, 6, 7, 5, -9, -8, -10, -6, -5, -7, -12, -11, -13, 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, -27, -26, -28, -24, -23, -25, -30, -29, -31, -18, -17, -19, -15, -14, -16, -21, -20, -22, -36

Offset: 0

Franklin T. Adams-Watters, Apr 05 2006

As the graph demonstrates, there are large discontinuities in the sequence between terms 3^i-1 and 3^i, and between terms 2*3^i-1 and 2*3^i. - N. J. A. Sloane, Jul 03 2016

			7 in base 3 is 21; changing the 2 to a (-1) gives (-1)*3+1 = -2, so a(7) = -2. I.e., the number of -2 according to the balanced ternary enumeration is 7, which can be obtained by replacing every -1 by 2 in the balanced ternary representation (or expansion) of -2, which is -1,1.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175; 2nd. ed. pp. 190-193.

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 729 terms from A. Karttunen)
Ken Levasseur, The Balanced Ternary Number System
Tilman Piesk, First 27 numbers with their ternary representation
N. J. A. Sloane and Brady Haran, Amazing Graphs II (including Star Wars), Numberphile video (2019)
Wikipedia, Balanced ternary

Cf. A117967, A117968, A001057, A004488, A134028, A274107, A059095, A005836, A289814, A244042.

Column k=1 of A319047.

f:= proc(n) local L,i;
   L:= subs(2=-1,convert(n,base,3));
   add(L[i]*3^(i-1),i=1..nops(L))
end proc:
map(f, [$0..100]);
# alternate:
N:= 100: # to get a(0) to a(N)
g:= 0:
for n from 1 to ceil(log[3](N+1)) do
g:= convert(series(3*subs(x=x^3,g)*(1+x+x^2)+x/(1+x+x^2),x,3^n+1),polynom);
od:
seq(coeff(g,x,j),j=0..N); # Robert Israel, Nov 17 2015
# third Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      3*a(iquo(n, 3, 'r'))+`if`(r=2, -1, r))
    end:
seq(a(n), n=0..3^4-1);  # Alois P. Heinz, Aug 14 2019

Map[FromDigits[#, 3] &, IntegerDigits[#, 3] /. 2 -> -1 & /@ Range@ 80] (* Michael De Vlieger, Nov 17 2015 *)

a(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n,3)), 'x), 'x, 3)
vector(73, i, a(i-1))  \\ Gheorghe Coserea, Nov 17 2015

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

a(0) = 0, a(3n) = 3a(n), a(3n+1) = 3a(n)+1, a(3n+2) = 3a(n)-1.
G.f. satisfies A(x) = 3*A(x^3)*(1+x+x^2) + x/(1+x+x^2). - corrected by Robert Israel, Nov 17 2015
A004488(n) = a(n)^{-1}(-a(n)). I.e., if a(n) <= 0, A004488(n) = A117967(-a(n)) and if a(n) > 0, A004488(n) = A117968(a(n)).
a(n) = n - 3 * A005836(A289814(n) + 1). - Andrey Zabolotskiy, Nov 11 2019

Name corrected by Andrey Zabolotskiy, Nov 10 2019

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.

A140267 Nonnegative integers in balanced ternary representation (with 2 standing for -1 digit).

Original entry on oeis.org

0, 1, 12, 10, 11, 122, 120, 121, 102, 100, 101, 112, 110, 111, 1222, 1220, 1221, 1202, 1200, 1201, 1212, 1210, 1211, 1022, 1020, 1021, 1002, 1000, 1001, 1012, 1010, 1011, 1122, 1120, 1121, 1102, 1100, 1101, 1112, 1110, 1111, 12222, 12220, 12221

Offset: 0

Antti Karttunen, May 19 2008, prompted by Eric Angelini's posting on SeqFan mailing list on Sep 15 2005

Sequence A117967 in ternary. (See there for more references.)
From Daniel Forgues, Mar 22 2010: (Start)
The balanced ternary digits {-1, 0, +1} (balanced trits) of a(n) are being represented by {2, 0, 1} respectively in this sequence.
The sign of a(n) is given by the sign of its leading trit.
The number k, k >= 0, of trailing "0"s of a(n) indicates that a(n) is divisible by 3^k.
a(n) is even/odd if it has an even/odd count of nonzero trits. (End)

			For example a(2) = 12, as 1*3 + -1*1 = 2. Similarly, a(19) = 1201, as 1*27 + -1*9 + 0*3 + 1*1 = 19.

Daniel Forgues, Table of n, a(n) for n = 0..100000
Jeff Connelly, Ternary Computing Testbed 3-Trit Computer Architecture, 2008. - _Daniel Forgues_, Mar 23 2010
Brian Hayes, Third Base, American Scientist, November-December 2001. - _Daniel Forgues_, Mar 23 2010
Ternary.info Forum, Balanced ternary arithmetics. - _Daniel Forgues_, Mar 23 2010

a(n) = A007089(A117967(n)). Cf. A140268.

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 a004488(a117968(n))
def a(n): return int("".join(map(str, digits(a117967(n), 3)[1:]))) # Indranil Ghosh, Jun 06 2017

Definition edited by Daniel Forgues, Mar 24 2010

A295882 Balanced ternary representation of the deficiency of n, A033879(n).

Original entry on oeis.org

1, 1, 5, 1, 4, 0, 15, 1, 17, 5, 10, 8, 12, 4, 15, 1, 52, 6, 45, 7, 10, 11, 49, 24, 46, 10, 53, 0, 28, 24, 30, 1, 45, 53, 49, 65, 36, 52, 49, 20, 40, 24, 159, 4, 12, 50, 154, 56, 161, 16, 30, 15, 142, 24, 41, 19, 43, 29, 139, 204, 150, 28, 49, 1, 154, 24, 147, 10, 159, 8, 106, 192, 99, 43, 29, 12, 139, 24, 87, 55

Offset: 1

Antti Karttunen, Dec 04 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A033879, A033880, A117966, A117967, A117968, A286449, A295881, A296071, A296072, A296073.

Cf. A000396 (gives the positions of zeros).

A117967(n) = if(n<=1,n,if(!(n%3),3*A117967(n/3),if(1==(n%3),1+3*A117967((n-1)/3),2+3*A117967((n+1)/3))));
A117968(n) = if(1==n,2,if(!(n%3),3*A117968(n/3),if(1==(n%3),2+3*A117968((n-1)/3),1+3*A117968((n+1)/3))));
A295882(n) = { my(x = (2*n)-sigma(n)); if(x >= 0,A117967(x),A117968(-x)); };

(define (A295882 n) (let ((x (A033879 n))) (if (>= x 0) (A117967 x) (A117968 (- x)))))

If A033879(n) >= 0, then a(n) = A117967(A033879(n)), otherwise a(n) = A117968(-A033879(n)).

For all n >= 1, A117966(a(n)) = A033879(n).

cp's OEIS Frontend

A140263 Permutation of nonnegative integers obtained by interleaving A117967 and A117968.

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A246207 Permutation of nonnegative integers: a(0) = 0, a(1) = 1, a(2n) = A117968(a(n)), a(2n+1) = A117967(1+a(n)).

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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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A004488 Tersum n + n.

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A117966 Balanced ternary enumeration (based on balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'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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