A117967 - OEIS

A117967

Positive part of inverse of A117966; write n in balanced ternary and then replace (-1)'s with 2's.

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

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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

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. - Antti Karttunen, May 19 2008

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

MATHEMATICA

a[n_] := Module[{d, i, m = n, r = 0}, For[i = 0, m > 0, i++, {m, d} = QuotientRemainder[m, 3]; If[d == 2, m++]; r = r + d*3^i]; r];

a /@ Range[0, 100] (* Jean-François Alcover, Jan 05 2021, after Alois P. Heinz *)

PROG

(Scheme)

(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))))

;; Antti Karttunen, May 19 2008

(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.