A005812 Weight of balanced ternary representation of n.

0, 1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 5, 4, 3

Offset: 0

Author

N. J. A. Sloane, Jeffrey Shallit

Keywords

• easy
• nonn
• nice

Comments

Weight of n means count of nonzero digits of n. - Daniel Forgues, Mar 24 2010

a(n) = A134022(n) + A134024(n) = A134021(n) - A134023(n).

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Daniel Forgues, Table of n, a(n) for n = 0..100000
• P. Flajolet and Lyle Ramshaw, A note on Gray code and odd-even merge, SIAM J. Comput. 9 (1980), 142-158.
• Michael Gilleland, Some Self-Similar Integer Sequences

Programs

Lisp

(defun btw (n) (if (= n 0) 0 (multiple-value-bind (q r) (round n 3) (+ (abs r) (btw q)))))

Mathematica

a[n_] := With[{q=Round[n/3]}, Abs[n-3q]+a[q]]; a[0]=0; Table[a[n], {n, 0, 105}](* Jean-François Alcover, Nov 25 2011, after Pari *)

PARI

a(n)=local(q); if(n<=0,0,q=round(n/3); abs(n-3*q)+a(q))

Python

def a(n):
    s=0
    x=0
    while n>0:
        x=n%3
        n//=3
        if x==2:
            x=-1
            n+=1
        if x!=0: s+=1
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 07 2017

Formula

a(3n)=a(n), a(3n+1)=a(n)+1, a(9n+2)=a(n)+2, a(9n+5)=a(3n+2)+1, a(9n+8)=a(3n+2).

a(n) = Sum_{k>0} floor(|2*sin(n*Pi/3^k)|). - Toshitaka Suzuki, Sep 10 2006

Extensions

Additional terms from Allan C. Wechsler