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A109973 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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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?

- Clark Kimberling, May 08 2011

LINKS

Table of n, a(n) for n=1..105.

MATHEMATICA

(* 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]

(* ...and so on.  By Peter J. C. Moses, Jun 03 2011 *)

CROSSREFS

Cf. A001155, A055168, A191654 (repeating AN iterates).

Sequence in context: A284532 A125585 A191860 * A220517 A271076 A087175

Adjacent sequences:  A109970 A109971 A109972 * A109974 A109975 A109976

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 06 2005

STATUS

approved

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Last modified February 18 13:07 EST 2018. Contains 299322 sequences. (Running on oeis4.)