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

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

0,3

Comments

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)

Links

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

Example

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.

Prog

(Python)
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

Crossrefs

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

Sequence in context: A216856 A040023 A109683 * A343530 A262965 A094450

Adjacent sequences: A140264 A140265 A140266 * A140268 A140269 A140270

Keyword

nonn,base

Author

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

Extensions

Definition edited by Daniel Forgues, Mar 24 2010

Status

approved