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.

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

Offset: 1

Author

Clark Kimberling, Jul 06 2005

Keywords

• nonn

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

Crossrefs

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

Programs

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 *)