A225869 Limiting sequence of a counting procedure.

2, 2, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 3, 1, 0, 0, 1, 2, 2, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 3, 1, 0, 0, 1, 2, 2, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 3, 1, 0, 0, 1, 2, 2, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 3, 1, 0, 0, 1, 2, 2, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 3, 1, 0, 0, 1, 2

Offset

1,1

Comments

Suppose that S = (x(1),..,x(h)) is a vector of nonnegative integers. Let m = max(S) and F(S) = (f(0),..,f(m)), where f(i) is the number of occurrences of i in S. Define F(0) = S and F(q) = F(F(q-1)) for q>=1. By Theorem 1 at A225660, the vector F(q) is eventually periodic with period 6.

Theorem 2. If S is not one of the ten vectors listed below, then F(q) = (2, 2) for some q, and the concatenation of F(q), F(q+1),... comprises the periodic sequence A225869. Seven of the exceptional vectors are indicated by (0) -> (1) -> (0,1) -> (1,1) -> (1,0,1) -> (1,2) -> (0,1,1) -> (1,2) -> ..., and the remaining three, by (2) -> (0,0,1) -> (2,1) -> (0,1,1). (The second appearances of (1,2) and (0,1,1) are not counted.)

A proof of Theorem 2 consists of easy (omitted) examinations of cases. Note, in particular, that A225869 is the limiting sequence for every S having more than 3 components.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Formula

The sequence is periodic with fundamental period 2, 2, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 3, 1, 0, 0, 1 .

Example

S = (6,1,0,5,5,3) -> (1,1,0,1,0,2,1) -> (2,4,1) -> (0,1,1,0,1) -> (2,3) -> (0,0,1,1) -> (2,2)* -> (0,0,2) -> (2,0,1) -> (1,1,1) -> (0,3) -> (1,0,0,1) -> (2,2).  The asterisk (*) shows where the limiting sequence A225869 begins.  The sequence is the concatenation of the repeating vectors starting with (2,2).

Mathematica

t[n_] := t[n] = Table[Count[t[n - 1], k], {k, 0, Max[t[n - 1]]}];
t[0] = {2, 2}; (* t[0] is the vector S*)
u = Table[t[n], {n, 0, 36}]  (* list of vectors F(q) *)
lst = Flatten[u]  (* A225869 as a sequence *)
PadRight[{}, 100, {2, 2, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 3, 1, 0, 0, 1}] (* Harvey P. Dale, Sep 10 2016 *)

Crossrefs

Cf. A225660

Sequence in context: A143432 A137677 A015818 * A039972 A344834 A344837

Adjacent sequences: A225866 A225867 A225868 * A225870 A225871 A225872

Keyword

nonn,easy

Author

Clark Kimberling, May 18 2013

Status

approved