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A117966 Balanced ternary enumeration (or balanced ternary representation) of integers; write n in ternary and then replace 2's with (-1)'s. 22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

REFERENCES

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

LINKS

A. Karttunen and Gheorghe Coserea, Table of n, a(n) for n = 0..54321(first 729 terms from A. Karttunen)

Ken Levasseur, The Balanced Ternary Number System

Tilman Piesk, First 27 numbers with their ternary representation

FORMULA

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)).

EXAMPLE

7 in base 3 is 21; changing the 2 to a (-1) gives (-1)*3+1 = -2, so a(7) = -2.

MAPLE

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

MATHEMATICA

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

PROG

(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

(PARI) 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

(Python)

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 xrange(101)] # Indranil Ghosh, Jun 06 2017

CROSSREFS

Cf. A117967, A117868, A001057, A004488, A134028, A274107.

Sequence in context: A316629 A254175 A088916 * A303932 A121891 A271590

Adjacent sequences:  A117963 A117964 A117965 * A117967 A117968 A117969

KEYWORD

base,sign

AUTHOR

Franklin T. Adams-Watters, Apr 05 2006

STATUS

approved

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Last modified December 16 09:34 EST 2018. Contains 318160 sequences. (Running on oeis4.)