A134021 Length of n in balanced ternary representation.

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

Offset

0,3

Comments

Shifted variant of A064099.

a(n) = A134022(n) + A134023(n) + A134024(n);

0 <= a(n) - A081604(n) <= 1;

a(A134025(n))=A081604(A134025(n)); a(A134026(n))=A081604(A134026(n))+1;

a(A134027(n)) = a(n); a(ABS(A134028(n))) <= a(n);

a(n) = A064099(n-1) for n>1.

n = Sum(A059095(A134421(n)-2-k)*3^k: 0<=k<a(n)), for n>0. - Reinhard Zumkeller, Oct 25 2007

a(n) = A005812(n) + A134023(n).

References

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Wikipedia, Balanced Ternary

Formula

For n>0: a(n) = ceiling(log(2*n+1)/log(3)).

Example

100=1*3^4+1*3^3-1*3^2+0*3^1+1*3^0: a(100)=|++-0+|=5;
200=1*3^5-1*3^4+1*3^3+1*3^2+1*3^1-1*3^0: a(200)=|+-+++-|=6;
300=1*3^5+1*3^4-1*3^3+0*3^2+1*3^1+0*3^0: a(300)=|++-0+0|=6.

Prog

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

Crossrefs

Sequence in context: A269024 A244160 A064099 * A330558 A237657 A244317

Adjacent sequences: A134018 A134019 A134020 * A134022 A134023 A134024

Keyword

nonn,base

Author

Reinhard Zumkeller, Oct 19 2007

Status

approved