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A117967 Positive part of inverse of A117966; write n in balanced ternary and then replace (-1)'s with 2's. 20
0, 1, 5, 3, 4, 17, 15, 16, 11, 9, 10, 14, 12, 13, 53, 51, 52, 47, 45, 46, 50, 48, 49, 35, 33, 34, 29, 27, 28, 32, 30, 31, 44, 42, 43, 38, 36, 37, 41, 39, 40, 161, 159, 160, 155, 153, 154, 158, 156, 157, 143, 141, 142, 137, 135, 136, 140, 138, 139, 152, 150, 151, 146 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..9841

Ken Levasseur, The Balanced Ternary Number System [broken link]

FORMULA

a(0) = 0, a(3n) = 3a(n), a(3n+1) = 3a(n)+1, a(3n-1) = 3a(n)+2.

If one adds this clause, then the function is defined on the whole Z: If n<0, then a(n) = A004488(a(-n)) (or equivalently: a(n) = A117968(-n)) and then it holds that a(A117966(n)) = n.

EXAMPLE

7 in balanced ternary is 1(-1)1, changing to 121 ternary is 16, so a(7)=16.

MAPLE

a:= proc(n) local d, i, m, r; m:=n; r:=0;

      for i from 0 while m>0 do

         d:= irem(m, 3, 'm');

         if d=2 then m:=m+1 fi;

         r:= r+d*3^i

      od; r

    end:

seq(a(n), n=0..100);  # Alois P. Heinz, May 11 2015

PROG

(Two alternative definitions in MIT/GNU Scheme, defined for whole Z:)

(define (A117967 z) (cond ((zero? z) 0) ((negative? z) (A004488 (A117967 (- z)))) (else (let* ((lp3 (expt 3 (A062153 z))) (np3 (* 3 lp3))) (if (< (* 2 z) np3) (+ lp3 (A117967 (- z lp3))) (+ np3 (A117967 (- z np3))))))))

(define (A117967v2 z) (cond ((zero? z) 0) ((negative? z) (A004488 (A117967v2 (- z)))) ((zero? (modulo z 3)) (* 3 (A117967v2 (/ z 3)))) ((= 1 (modulo z 3)) (+ (* 3 (A117967v2 (/ (- z 1) 3))) 1)) (else (+ (* 3 (A117967v2 (/ (+ z 1) 3))) 2))))

(Python)

from sympy.ntheory.factor_ import digits

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

def a117968(n):

    if n==1: return 2

    if n%3==0: return 3*a117968(n/3)

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

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

def a(n): return 0 if n==0 else a004488(a117968(n)) # Indranil Ghosh, Jun 06 2017

CROSSREFS

Cf. A117966. a(n) = A004488(A117968(n)). Bisection of A140263. A140267 gives the same sequence in ternary.

Sequence in context: A086308 A229943 A198132 * A068116 A275836 A019172

Adjacent sequences:  A117964 A117965 A117966 * A117968 A117969 A117970

KEYWORD

base,nonn,look

AUTHOR

Franklin T. Adams-Watters, Apr 05 2006

EXTENSIONS

Antti Karttunen, May 19 2008, extended the formula's domain to all integers and added the Scheme code.

STATUS

approved

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Last modified January 18 01:36 EST 2019. Contains 319260 sequences. (Running on oeis4.)