%I #7 Jul 16 2020 02:35:06
%S 1,11,21,1112,3112,211213,312213,212223,1110213,101011213,201111213,
%T 101112213,101112213,101112213,101112213,101112213,101112213,
%U 101112213,101112213,101112213,101112213,101112213,101112213,101112213,101112213
%N Summarize the previous term in base 4 (in increasing order).
%C Let a(1)=1. Describing a(1) as "one 1" again gives a(2)=11 (same digit string as A005151 and similar sequences). Likewise, a(3) through a(8) have the same digit strings as the corresponding terms of A005151, but describing a(8) as "one 1, four 2s, one 3" gives a(9)=1110213 when the frequency of digit occurrence is written in base 4 and followed by the digit counted.
%H Onno M. Cain, Sela T. Enin, <a href="https://arxiv.org/abs/2004.00209">Inventory Loops (i.e. Counting Sequences) have Pre-period 2 max S_1 + 60</a>, arXiv:2004.00209 [math.NT], 2020.
%F a(n) = 101112213 for all n >= 12 (see example).
%e Summarizing a(12) = 101112213 in increasing digit order, there are "one 0, five 1's, two 2s, one 3", so concatenating 1 0 11 1 2 2 1 3 gives a(13) = 101112213 (=a(14)=a(15)=...).
%t Nest[Append[#, FromDigits[Flatten@ Map[IntegerDigits[#, 4] & /@ Reverse@ # &, Tally@ Sort@ IntegerDigits@ #[[-1]] ] ]] &, {1}, 24] (* _Michael De Vlieger_, Jul 15 2020 *)
%Y Cf. A098153 (binary), A098154 (ternary), A005151 (decimal and digit strings for all other bases b >= 5).
%K base,easy,nonn
%O 1,2
%A _Rick L. Shepherd_, Aug 29 2004