A292524 Interpret the values of the Moebius function mu(k) for k = 1 to n as a balanced ternary number.

0, 1, 2, 5, 15, 44, 133, 398, 1194, 3582, 10747, 32240, 96720, 290159, 870478, 2611435, 7834305, 23502914, 70508742, 211526225, 634578675, 1903736026, 5711208079, 17133624236, 51400872708, 154202618124, 462607854373, 1387823563119, 4163470689357

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

Cf. A008683, A238271, A292779, A344432, A344433.

Sequence in context: A094176 A084086 A307259 * A071742 A035355 A220873

Adjacent sequences: A292521 A292522 A292523 * A292525 A292526 A292527

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