OFFSET
0,3
COMMENTS
Balanced ternary is much like regular ternary, but with the crucial difference of using the digit -1 instead of the digit 2. Then some powers of 3 are added, others are subtracted.
If mu(n) = 0, then a(n) is a multiple of 3, specifically, it is thrice a(n - 1). Otherwise, a(n) is not a multiple of 3.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2097
Wikipedia, Balanced ternary
FORMULA
a(n) = Sum_{k = 1..n} mu(k) 3^(n - k).
a(n) = 3 * a(n-1) + mu(n) for n > 0. - Alois P. Heinz, Oct 13 2017
a(n) ~ A238271 * 3^n. - Vaclav Kotesovec, May 19 2021
EXAMPLE
mu(1) = 1, so a(1) = 1 * 3^0 = 1.
mu(2) = -1, so a(2) = 1 * 3^1 + -1 * 3^0 = 3 - 1 = 2.
mu(3) = -1, so a(3) = 1 * 3^2 + -1 * 3^1 + -1 * 3^0 = 9 - 3 - 1 = 5.
mu(4) = 0, so a(4) = 1 * 3^3 + -1 * 3^2 + -1 * 3^1 + 0 * 3^0 = 27 - 9 - 3 + 0 = 15.
MAPLE
a:= proc(n) option remember; `if`(n=0, 0,
a(n-1)*3+numtheory[mobius](n))
end:
seq(a(n), n=0..33); # Alois P. Heinz, Oct 13 2017
MATHEMATICA
Table[Plus@@(3^Range[n - 1, 0, -1] MoebiusMu[Range[n]]), {n, 50}]
PROG
(PARI) a(n) = sum(k=1, n, moebius(k)*3^(n-k)); \\ Michel Marcus, Oct 01 2017
(PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, moebius(k)*x^k)/(1-3*x))) \\ Seiichi Manyama, May 19 2021
(PARI) a(n) = if(n==0, 0, 3*a(n-1)+moebius(n)); \\ Seiichi Manyama, May 19 2021
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
Alonso del Arte, Sep 18 2017
EXTENSIONS
a(0)=0 prepended by Alois P. Heinz, Oct 13 2017
STATUS
approved