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A109973
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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.
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1
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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, 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, 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
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OFFSET
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1,4
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COMMENTS
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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.
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.
Third segment (from initial 2): 01132133.
Fourth segment (from initial 3): 0113223341.
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.
These observations prompt questions: (1) what initial segments generate fixed points? (2) do segments eventually occur periodically, regardless of the choice of initial segment?
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LINKS
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MATHEMATICA
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(* Program computes the NA segment starting with 0 *)
nounAdjective[s_] := Flatten@Transpose@({#1, (Count[s, #1] &) /@ #1} &)[Range[0, Max[s]]];
NestList[nounAdjective[#1] &, nounAdjective[{0}], 7]
(* Next program: the NA segment starting with 1 *)
nounAdjective[s_] := Flatten@Transpose@({#1, (Count[s, #1] &) /@ #1} &)[Range[0, Max[s]]];
NestList[nounAdjective[#1] &, nounAdjective[{1}], 7]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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