A350651 a(n) is the size of the orbit of n under repeated application of A350229 (the sum of a number and its balanced ternary digits).

1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 3, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 3, 2, 7, 1, 7, 6, 6, 5, 4, 4, 4, 3, 5, 3, 4, 3, 4, 1, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 3, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 3, 2, 6, 1

Offset

0,2

Comments

This sequence is well defined:

- for any w such that 3^w > 2*w + 2:

- let M(w) (resp. m(w)) be the number whose balanced ternary expansion

starts with "1T"

followed by w T's

followed by w 1's (resp. w T's),

- for any u in the interval m(w)..M(w), A350229(u) <= u <= M(w),

- also for any v < m(k), A350229(v) < m(k) + 2*w + 2 <= M(w),

- and the orbit of any number <= M(w) will be trapped in the interval 0..M(w),

- hence for any number n, we can find an appropriate w, so the orbit of n is bounded and eventually periodic (with a finite size), QED.

This sequence is unbounded.

Links

Table of n, a(n) for n=0..86.

Formula

a(n) = #{ A350229^k(n), k >= 0 } (where f^k corresponds to the k-th iterate of f).

Example

For n = 9:
- the orbit of 9 contains the following values:
  k  v   bter(v)  ds(v)
  -  --  -------  -----
  0   9      100      1
  1  10      101      2
  2  12      110      2
  3  14     1TTT     -2
  4  12      110      2
- so a(9) = #{ 9, 10, 12, 14 } = 4.

Mathematica

f[n_] := n + Total[If[First@ # == 0, Rest@ #, #] &[Prepend[IntegerDigits[n, 3], 0] //. {x___, y_, k_ /; k > 1, z___} :> {x, y + 1, k - 3, z}]]; Array[-1 + Length@ NestWhileList[f, #, UnsameQ, All] &, 105, 0] (* Michael De Vlieger, Jan 15 2022 *)

Prog

(PARI) b(n) = my (v=n, d); while (n, n=(n-d=[0, 1, -1][1+n%3])/3; v+=d); v
a(n) = my (s=[]); while (!setsearch(s, n), s=setunion(s, [n]); n=b(n)); #s

Crossrefs

Cf. A065363, A350229.

Sequence in context: A260452 A005679 A232927 * A275832 A237839 A101608

Adjacent sequences: A350648 A350649 A350650 * A350652 A350653 A350654

Keyword

nonn,base

Author

Rémy Sigrist, Jan 09 2022

Status

approved