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Describe all the numbers already used (sorted into increasing order - not splitting numbers up into their digits).
15

%I #18 Jan 11 2021 18:14:39

%S 1,11,31,4113,612314,8112332416,1113253342618,131528344153628111,

%T 1617210364354648211113,181921239445661758110311213116,

%U 2211121431146586276829210411112313216118

%N Describe all the numbers already used (sorted into increasing order - not splitting numbers up into their digits).

%H Reinhard Zumkeller, <a href="/A060857/b060857.txt">Table of n, a(n) for n = 0..60</a>

%H Onno M. Cain and 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.

%e One; one one; three ones; four ones, one three; six ones, two threes, one four; eight ones, one two, three threes, two fours, one six; eleven ones, three twos, five threes, three fours, two sixes, one eight; thirteen [note not 15] ones, five twos, eight threes, four fours, one five, three sixes, two eights, one eleven [note than numbers >9 are preserved as wholes rather than as a collection of digits]; etc.

%t FromDigits /@ Nest[Append[#, Flatten@ Map[Reverse, Tally@ Sort@ Flatten@ # ] ] &, {{1}}, 10] (* _Michael De Vlieger_, Jul 15 2020 *)

%o (Haskell)

%o import Data.List (group, sort, transpose)

%o a060857 n = a060857_list !! n

%o a060857_list = 1 : f [1] :: [Integer] where

%o f xs = (read $ concatMap show ys) : f (xs ++ ys) where

%o ys = concat $ transpose [map length zss, map head zss]

%o zss = group $ sort xs

%o -- _Reinhard Zumkeller_, Jan 25 2014

%o (Python)

%o def summarize_lst(lst):

%o ans = []

%o for d in sorted(set(lst)): ans += [lst.count(d), d]

%o return ans

%o def aupton(nn):

%o alst, arunninglst = [1], [1]

%o for n in range(nn):

%o nxt_lst = summarize_lst(arunninglst)

%o arunninglst += nxt_lst

%o alst.append(int("".join(map(str, nxt_lst))))

%o return alst

%o print(aupton(10)) # _Michael S. Branicky_, Jan 11 2021

%Y This is a combination of methods used in A005151 and A045982. The first word of each term (the number of ones used earlier) seems to be equal to A030711 and A030761.

%K base,nice,nonn

%O 0,2

%A _Henry Bottomley_, May 03 2001