A140267 - OEIS

%I #14 Jun 06 2017 19:45:42

%S 0,1,12,10,11,122,120,121,102,100,101,112,110,111,1222,1220,1221,1202,

%T 1200,1201,1212,1210,1211,1022,1020,1021,1002,1000,1001,1012,1010,

%U 1011,1122,1120,1121,1102,1100,1101,1112,1110,1111,12222,12220,12221

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

%C Sequence A117967 in ternary. (See there for more references.)

%C From _Daniel Forgues_, Mar 22 2010: (Start)

%C The balanced ternary digits {-1, 0, +1} (balanced trits) of a(n) are being represented by {2, 0, 1} respectively in this sequence.

%C The sign of a(n) is given by the sign of its leading trit.

%C The number k, k >= 0, of trailing "0"s of a(n) indicates that a(n) is divisible by 3^k.

%C a(n) is even/odd if it has an even/odd count of nonzero trits. (End)

%H Daniel Forgues, <a href="/A140267/b140267.txt">Table of n, a(n) for n = 0..100000</a>

%H Jeff Connelly, <a href="http://xyzzy.freeshell.org/trinary/CPE%20Report%20-%20Ternary%20Computing%20Testbed%20-%20RC6a.pdf">Ternary Computing Testbed 3-Trit Computer Architecture</a>, 2008. - _Daniel Forgues_, Mar 23 2010

%H Brian Hayes, <a href="http://www.americanscientist.org/issues/pub/third-base/1">Third Base</a>, American Scientist, November-December 2001. - _Daniel Forgues_, Mar 23 2010

%H Ternary.info Forum, <a href="http://ternary.info/modules/newbb/viewtopic.php?topic_id=116&amp;forum=10">Balanced ternary arithmetics</a>. - _Daniel Forgues_, Mar 23 2010

%e 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.

%o (Python)

%o from sympy.ntheory.factor_ import digits

%o def a004488(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3)

%o def a117968(n):

%o     if n==1: return 2

%o     if n%3==0: return 3*a117968(n/3)

%o     elif n%3==1: return 3*a117968((n - 1)/3) + 2

%o     else: return 3*a117968((n + 1)/3) + 1

%o def a117967(n): return 0 if n==0 else a004488(a117968(n))

%o def a(n): return int("".join(map(str, digits(a117967(n), 3)[1:]))) # _Indranil Ghosh_, Jun 06 2017

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

%K nonn,base

%O 0,3

%A _Antti Karttunen_, May 19 2008, prompted by _Eric Angelini_'s posting on SeqFan mailing list on Sep 15 2005.

%E Definition edited by _Daniel Forgues_, Mar 24 2010