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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.

%I A117966 #87 Mar 10 2021 07:23:14
%S A117966 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,
%T A117966 -11,-13,27,28,26,30,31,29,24,25,23,36,37,35,39,40,38,33,34,32,18,19,
%U A117966 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
%N A117966 Balanced ternary enumeration (based on balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'s.
%C A117966 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
%D A117966 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175; 2nd. ed. pp. 190-193.
%H A117966 Gheorghe Coserea, <a href="/A117966/b117966.txt">Table of n, a(n) for n = 0..59048</a> (first 729 terms from A. Karttunen)
%H A117966 Ken Levasseur, <a href="http://discretemath.org/ternary_number_system.html">The Balanced Ternary Number System</a>
%H A117966 Tilman Piesk, <a href="http://commons.wikimedia.org/wiki/File:Balanced_ternary_A117966.svg">First 27 numbers with their ternary representation</a>
%H A117966 N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=o8c4uYnnNnc">Amazing Graphs II (including Star Wars)</a>, Numberphile video (2019)
%H A117966 Wikipedia, <a href="https://en.wikipedia.org/wiki/Balanced_ternary">Balanced ternary</a>
%F A117966 a(0) = 0, a(3n) = 3a(n), a(3n+1) = 3a(n)+1, a(3n+2) = 3a(n)-1.
%F A117966 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
%F A117966 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)).
%F A117966 a(n) = n - 3 * A005836(A289814(n) + 1). - _Andrey Zabolotskiy_, Nov 11 2019
%e A117966 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.
%p A117966 f:= proc(n) local L,i;
%p A117966    L:= subs(2=-1,convert(n,base,3));
%p A117966    add(L[i]*3^(i-1),i=1..nops(L))
%p A117966 end proc:
%p A117966 map(f, [$0..100]);
%p A117966 # alternate:
%p A117966 N:= 100: # to get a(0) to a(N)
%p A117966 g:= 0:
%p A117966 for n from 1 to ceil(log[3](N+1)) do
%p A117966 g:= convert(series(3*subs(x=x^3,g)*(1+x+x^2)+x/(1+x+x^2),x,3^n+1),polynom);
%p A117966 od:
%p A117966 seq(coeff(g,x,j),j=0..N); # _Robert Israel_, Nov 17 2015
%p A117966 # third Maple program:
%p A117966 a:= proc(n) option remember; `if`(n=0, 0,
%p A117966       3*a(iquo(n, 3, 'r'))+`if`(r=2, -1, r))
%p A117966     end:
%p A117966 seq(a(n), n=0..3^4-1);  # _Alois P. Heinz_, Aug 14 2019
%t A117966 Map[FromDigits[#, 3] &, IntegerDigits[#, 3] /. 2 -> -1 & /@ Range@ 80] (* _Michael De Vlieger_, Nov 17 2015 *)
%o A117966 (MIT/GNU Scheme:) (define (A117966 n) (let loop ((z 0) (i 0) (n n)) (if (zero? n) z (loop (+ z (* (expt 3 i) (if (= 2 (modulo n 3)) -1 (modulo n 3)))) (1+ i) (floor->exact (/ n 3)))))) -- _Antti Karttunen_, May 19 2008
%o A117966 (PARI) a(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n,3)), 'x), 'x, 3)
%o A117966 vector(73, i, a(i-1))  \\ _Gheorghe Coserea_, Nov 17 2015
%o A117966 (Python)
%o A117966 def a(n):
%o A117966     if n==0: return 0
%o A117966     if n%3==0: return 3*a(n//3)
%o A117966     elif n%3==1: return 3*a((n - 1)//3) + 1
%o A117966     else: return 3*a((n - 2)//3) - 1
%o A117966 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 06 2017
%Y A117966 Cf. A117967, A117968, A001057, A004488, A134028, A274107, A059095, A005836, A289814, A244042.
%Y A117966 Column k=1 of A319047.
%K A117966 base,sign,look
%O A117966 0,4
%A A117966 _Franklin T. Adams-Watters_, Apr 05 2006
%E A117966 Name corrected by _Andrey Zabolotskiy_, Nov 10 2019