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

%I A140267 #16 Jul 09 2025 04:29:30
%S A140267 0,1,12,10,11,122,120,121,102,100,101,112,110,111,1222,1220,1221,1202,
%T A140267 1200,1201,1212,1210,1211,1022,1020,1021,1002,1000,1001,1012,1010,
%U A140267 1011,1122,1120,1121,1102,1100,1101,1112,1110,1111,12222,12220,12221
%N A140267 Nonnegative integers in balanced ternary representation (with 2 standing for -1 digit).
%C A140267 Sequence A117967 in ternary. (See there for more references.)
%C A140267 From _Daniel Forgues_, Mar 22 2010: (Start)
%C A140267 The balanced ternary digits {-1, 0, +1} (balanced trits) of a(n) are being represented by {2, 0, 1} respectively in this sequence.
%C A140267 The sign of a(n) is given by the sign of its leading trit.
%C A140267 The number k, k >= 0, of trailing "0"s of a(n) indicates that a(n) is divisible by 3^k.
%C A140267 a(n) is even/odd if it has an even/odd count of nonzero trits. (End)
%H A140267 Daniel Forgues, <a href="/A140267/b140267.txt">Table of n, a(n) for n = 0..100000</a>
%H A140267 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 A140267 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 A140267 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 A140267 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 A140267 (Python)
%o A140267 from sympy.ntheory.factor_ import digits
%o A140267 def a004488(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3)
%o A140267 def a117968(n):
%o A140267     if n==1: return 2
%o A140267     if n%3==0: return 3*a117968(n/3)
%o A140267     elif n%3==1: return 3*a117968((n - 1)/3) + 2
%o A140267     else: return 3*a117968((n + 1)/3) + 1
%o A140267 def a117967(n): return 0 if n==0 else a004488(a117968(n))
%o A140267 def a(n): return int("".join(map(str, digits(a117967(n), 3)[1:]))) # _Indranil Ghosh_, Jun 06 2017
%Y A140267 a(n) = A007089(A117967(n)). Cf. A140268.
%K A140267 nonn,base
%O A140267 0,3
%A A140267 _Antti Karttunen_, May 19 2008, prompted by _Eric Angelini_'s posting on SeqFan mailing list on Sep 15 2005
%E A140267 Definition edited by _Daniel Forgues_, Mar 24 2010