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

Author

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

Keywords

• nonn
• base

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)

Examples

			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.
		

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

Crossrefs

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

Programs

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

Extensions

Definition edited by Daniel Forgues, Mar 24 2010