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%I #9 Feb 13 2014 00:47:27
%S 0,1,1,2,2,3,3,2,0,1,1,3,2,1,3,3,0,1,1,3,2,1,3,3,0,1,1,3,2,2,3,3,4,1,
%T 0,1,1,4,2,2,3,2,4,2,5,1,0,1,1,5,2,2,3,2,4,1,5,2,6,1,0,1,1,5,2,4,3,1,
%U 4,1,5,1,6,2,7,1,0,1,1,6,2,4,3,1,4,1,5,1,6,1,7,2,8,1,0,1,1,7,2,4,3,1,4,1,5
%N First repeating NA iterates. The NA (Noun-Adjective) function of a finite sequence s of nonnegative integers is the finite sequence 0a1b2c...mz, where a = #0's in s, b = #1's in s, ..., z = #m's in s, m = greatest term in s.
%C This is a concatenation of finite segments. The first segment is 01122332, obtained by writing the NA iterates of 0 until repetition occurs: 0, 01, 0111, 0113, 01122031, 02132231, 01122332, after which 01122332 repeats. It helps to speak your way through: write 0 and say 0 one time - that's 01; then say 0 one time and 1 one time - that's 0111; then say 0 one time and 1 3 times, and so on, until reaching the repeating segment 01122332. This segment is a fixed point of the NA function.
%C The second segment is obtained by writing the NA iterates of 1 until repetition occurs: 1, 0011, 0212, 011122, 011322, 01122231, 01132331, 01132133, after which 01132133 repeats, so that the second segment is 01132133.
%C Third segment (from initial 2): 01132133.
%C Fourth segment (from initial 3): 0113223341.
%C Fifth segment (from initial 4): 011422324251. Here, for the first time, the repetition does not occur immediately after the first occurrence. Indeed, iteration never reaches a fixed point of the NA function. Instead, the iterates oscillate between 011422324251 and 011324314251.
%C These observations prompt questions: (1) what initial segments generate fixed points? (2) do segments eventually occur periodically, regardless of the choice of initial segment?
%C - _Clark Kimberling_, May 08 2011
%t (* Program computes the NA segment starting with 0 *)
%t nounAdjective[s_] := Flatten@Transpose@({#1, (Count[s, #1] &) /@ #1} &)[Range[0, Max[s]]];
%t NestList[nounAdjective[#1] &, nounAdjective[{0}], 7]
%t (* Next program: the NA segment starting with 1 *)
%t nounAdjective[s_] := Flatten@Transpose@({#1, (Count[s, #1] &) /@ #1} &)[Range[0, Max[s]]];
%t NestList[nounAdjective[#1] &, nounAdjective[{1}], 7]
%t (* ...and so on. By _Peter J. C. Moses_, Jun 03 2011 *)
%Y Cf. A001155, A055168, A191654 (repeating AN iterates).
%K nonn
%O 1,4
%A _Clark Kimberling_, Jul 06 2005
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