A274107 -id:A274107 - cp's OEIS frontend

A117966 Balanced ternary enumeration (based on balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'s.

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, -11, -13, 27, 28, 26, 30, 31, 29, 24, 25, 23, 36, 37, 35, 39, 40, 38, 33, 34, 32, 18, 19, 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

Offset: 0

Franklin T. Adams-Watters, Apr 05 2006

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

			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.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, pp. 173-175; 2nd. ed. pp. 190-193.

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 729 terms from A. Karttunen)
Ken Levasseur, The Balanced Ternary Number System
Tilman Piesk, First 27 numbers with their ternary representation
N. J. A. Sloane and Brady Haran, Amazing Graphs II (including Star Wars), Numberphile video (2019)
Wikipedia, Balanced ternary

Cf. A117967, A117968, A001057, A004488, A134028, A274107, A059095, A005836, A289814, A244042.

Column k=1 of A319047.

f:= proc(n) local L,i;
   L:= subs(2=-1,convert(n,base,3));
   add(L[i]*3^(i-1),i=1..nops(L))
end proc:
map(f, [$0..100]);
# alternate:
N:= 100: # to get a(0) to a(N)
g:= 0:
for n from 1 to ceil(log[3](N+1)) do
g:= convert(series(3*subs(x=x^3,g)*(1+x+x^2)+x/(1+x+x^2),x,3^n+1),polynom);
od:
seq(coeff(g,x,j),j=0..N); # Robert Israel, Nov 17 2015
# third Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      3*a(iquo(n, 3, 'r'))+`if`(r=2, -1, r))
    end:
seq(a(n), n=0..3^4-1);  # Alois P. Heinz, Aug 14 2019

Map[FromDigits[#, 3] &, IntegerDigits[#, 3] /. 2 -> -1 & /@ Range@ 80] (* Michael De Vlieger, Nov 17 2015 *)

a(n) = subst(Pol(apply(x->if(x == 2, -1, x), digits(n,3)), 'x), 'x, 3)
vector(73, i, a(i-1))  \\ Gheorghe Coserea, Nov 17 2015

def a(n):
    if n==0: return 0
    if n%3==0: return 3*a(n//3)
    elif n%3==1: return 3*a((n - 1)//3) + 1
    else: return 3*a((n - 2)//3) - 1
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 06 2017

a(0) = 0, a(3n) = 3a(n), a(3n+1) = 3a(n)+1, a(3n+2) = 3a(n)-1.
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
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)).
a(n) = n - 3 * A005836(A289814(n) + 1). - Andrey Zabolotskiy, Nov 11 2019

Name corrected by Andrey Zabolotskiy, Nov 10 2019

A134028 Reversal of n in balanced ternary representation.

Original entry on oeis.org

0, 1, -2, 1, 4, -11, -2, 7, -8, 1, 10, -5, 4, 13, -38, -11, 16, -29, -2, 25, -20, 7, 34, -35, -8, 19, -26, 1, 28, -17, 10, 37, -32, -5, 22, -23, 4, 31, -14, 13, 40, -119, -38, 43, -92, -11, 70, -65, 16, 97, -110, -29, 52, -83, -2, 79, -56, 25, 106, -101, -20, 61, -74, 7, 88, -47, 34, 115, -116, -35, 46, -89, -8, 73, -62, 19, 100

Offset: 0

Reinhard Zumkeller, Oct 19 2007

As the graph demonstrates, the sequence makes large negative steps at terms (3^i+1)/2. These steps divide the graph into conspicuous blocks. - N. J. A. Sloane, Jul 03 2016

			20 = 1*3^3 - 1*3^2 + 1*3^1 - 1*3^0 == '+-+-'
=> a(20) = -1*3^3 + 1*3^2 - 1*3^1 + 1*3^0 = -20;
21 = 1*3^3 - 1*3^2 + 1*3^1 + 0*3^0 == '+-+0'
=> a(21) = 0*3^3 + 1*3^2 - 1*3^1 + 1*3^0 = 7;
22 = 1*3^3 - 1*3^2 + 1*3^1 + 1*3^0 == '+-++'
=> a(22) = 1*3^3 + 1*3^2 - 1*3^1 + 1*3^0 = 34;
23 = 1*3^3 + 0*3^2 - 1*3^1 - 1*3^0 == '+0--'
=> a(23) = -1*3^3 - 1*3^2 + 0*3^1 + 1*3^0 = -35;
24 = 1*3^3 + 0*3^2 - 1*3^1 + 0*3^0 == '+0-0'
=> a(24) = 0*3^3 - 1*3^2 + 0*3^1 + 1*3^0 = -8;
25 = 1*3^3 + 0*3^2 - 1*3^1 + 1*3^0 == '+0-+'
=> a(25) = 1*3^3 - 1*3^2 + 0*3^1 + 1*3^0 = 19.

D. E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, Vol 2, pp 173-175.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Reversal
Wikipedia, Balanced Ternary

Cf. A117966 (balanced ternary representation), A030102, A134021, A274107.

A134027 gives the numbers whose balanced ternary representation is palindromic.

def a(n):
    if n==0: return 0
    s=[]
    x=0
    while n>0:
        x=n%3
        n=n//3
        if x==2:
            x=-1
            n+=1
        s.append(x)
    l=s[::-1]
    return sum(l[i]*3**i for i in range(len(l)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 10 2017

a(A134027(n)) = A134027(n);

A134021(ABS(a(n))) <= A134021(n).

cp's OEIS Frontend

A117966 Balanced ternary enumeration (based on balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'s.

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