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%I #7 Sep 10 2016 13:59:23
%S 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,
%T 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,
%U 2,2,0,0,2,2,0,1,1,1,1,0,3,1,0,0,1,2
%N Limiting sequence of a counting procedure.
%C 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.
%C 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.)
%C 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.
%H Clark Kimberling, <a href="/A225869/b225869.txt">Table of n, a(n) for n = 1..1000</a>
%F The sequence is periodic with fundamental period 2, 2, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 3, 1, 0, 0, 1 .
%e 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).
%t t[n_] := t[n] = Table[Count[t[n - 1], k], {k, 0, Max[t[n - 1]]}];
%t t[0] = {2, 2}; (* t[0] is the vector S*)
%t u = Table[t[n], {n, 0, 36}] (* list of vectors F(q) *)
%t lst = Flatten[u] (* A225869 as a sequence *)
%t 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 *)
%Y Cf. A225660
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, May 18 2013
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