A292779 Interpret the values of the Moebius function mu(k) for k = n to 1 as a balanced ternary number.

1, -2, -11, -11, -92, 151, -578, -578, -578, 19105, -39944, -39944, -571385, 1022938, 5805907, 5805907, -37240814, -37240814, -424661303, -424661303, 3062123098, 13522476301, -17858583308, -17858583308, -17858583308, 829430026135, 829430026135, 829430026135

Offset

1,2

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.

Since the least significant digit is always 1, a(n) is never a multiple of 3.

If mu(n) = 0, then a(n) is the same as a(n - 1).

Run lengths are given by A076259. - Andrey Zabolotskiy, Oct 13 2017

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Wikipedia, Balanced ternary

Formula

a(n) = Sum_{k = 1 .. n} mu(k) 3^(k - 1).

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 = -11.
mu(4) = 0, so a(4) = 0 * 3^3 + -1 * 3^2 + -1 * 3^1 + 1 * 3^0 = -9 - 3 + 1 = -11.

Maple

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1)+3^(n-1)*numtheory[mobius](n))
    end:
seq(a(n), n=1..33);  # Alois P. Heinz, Oct 13 2017

Mathematica

Table[3^Range[0, n - 1].MoebiusMu[Range[n]], {n, 50}]

Prog

(PARI) a(n) = sum(k=1, n, moebius(k)*3^(k-1)); \\ Michel Marcus, Oct 01 2017

Crossrefs

Cf. A008683, A127513, A292524.

Sequence in context: A265545 A153705 A290394 * A245521 A275617 A090009

Adjacent sequences: A292776 A292777 A292778 * A292780 A292781 A292782

Keyword

easy,sign,base

Author

Alonso del Arte, Sep 22 2017

Status

approved