%I
%S 1,0,2,1,3,2,2,3,1,0,3,1,1,2,3,3,1,0,3,1,1,2,3,3,1,0,3,1,2,2,3,3,1,4,
%T 1,0,4,1,2,2,2,3,2,4,1,5,1,0,5,1,2,2,2,3,1,4,2,5,1,6,1,0,5,1,4,2,1,3,
%U 1,4,1,5,2,6,1,7,1,0,6,1,4,2,1,3,1,4,1,5,1,6,2,7,1,8
%N First repeating AN iterates. The AN (AdjectivebeforeNoun) function of a finite sequence s of nonnegative integers is the finite sequence a,0,b,1,c,2,...m,z, where a=#0's in s, b=#1's in s,..., m=#z's in s, where m is the greatest term in s.
%C This is a concatenation of finite segments. The first segment is 10213223, obtained by writing AN iterates starting with 0 until repetition occurs: 0, 10, 1011, 1031, 10210213, 20312213, 10213223, 10213223. It may help to speak your way along: write 0 and say one 0  that's 10; then say one 0 and one 1  that's 1011; and so on, until reaching the repeating segment 10213223. This segment is a fixed point of the AN function.
%C The second segment arises in the same way starting with 1, and likewise for further segments. The resulting segments concatenate to form A191654 in the same manner that NA segments form A109973. Indeed, A191654 can be easily read from A109973 by reversing pairs of terms. Thus, the open questions at A109973 apply also to A191654.
%t (* Program computes the AN segment starting with 0. *)
%t adjectiveNoun[s_] := Flatten@Transpose@({(Count[s, #1] &) /@ #1, #1} &)[Range[0, Max[s]]];
%t NestList[adjectiveNoun[#1] &, adjectiveNoun[{0}], 7]
%t (* Next program, the AN segment starting with 1. *)
%t adjectiveNoun[s_] := Flatten@Transpose@({(Count[s, #1] &) /@ #1, #1} &)[Range[0, Max[s]]];
%t NestList[adjectiveNoun[#1] &, adjectiveNoun[{1}], 7]
%t (* ...and so on. By _Peter J. C. Moses_, Jun 03 2011 *)
%Y Cf. A109973.
%K nonn
%O 1,3
%A _Clark Kimberling_, Jun 10 2011
