A327246 - OEIS

%I #58 Dec 09 2023 01:43:14

%S 0,11,21,22,23,24,25,26,27,28,29,210,31,211,32,212,33,213,34,214,35,

%T 215,36,216,37,217,38,218,39,219,310,311,312,313,314,315,316,317,318,

%U 319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335

%N a(1) = 0; build a(n+1) by concatenating 2 strings: first, the number of digits of a(n), second, the number of terms with same number of digits as a(n) up to a(n).

%C Empirical observations:

%C The sequence is not sensitive to the initial value; sooner or later the sequence arrives at 21 and has the same trajectory thereafter.

%C Certain numbers cannot appear more than once. An example is 210, since it signifies a(n) was the 10th instance of a 2-digit number. We cannot ever say that a 21-digit number has appeared 0 times. However, 211 may signify a(n) was the 11th instance of a 2-digit number, or the first instance of a 21-digit number. It is difficult to ascertain the structure of repetitions, when they occur, etc.

%H Michael De Vlieger, <a href="/A327246/b327246.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A327246/a327246.png">Log log scatterplot of a(n)</a> for n = 1..2^20.

%e a(1) = 0.

%e a(2) = 11 since 0 is the first instance of a 1-digit number in the sequence,

%e a(3) = 21 since 11 is the first instance of a 2-digit number,

%e a(4) = 22 since 21 is the second instance of a 2-digit number, ...

%e a(12) = 210 since 29 is the tenth instance of a 2-digit number,

%e a(13) = 31 since 210 is the first instance of a 3-digit number, etc.

%e a(1) can be any positive integer number between 0-9, the sequence will not have further change. The overall sequence is defined by the number of digits of a(1).

%e For example if a(1) were 1000000000 or 9999999999, then we would have a(2)=101, a(3)=31, a(4)=21, a(5)=22, ... where a(1) has 10 digits so a(2) would start with 10 and ends with 1, etc.

%t iL[n_] := IntegerLength[n] + Boole[n==0]; a[1]=0; a[n_] := a[n] = Block[{t = iL@ a[n-1]}, FromDigits[ Join @@ IntegerDigits@ {t, Count[iL /@ Array[a, n - 1], t]}]]; Array[a, 56] (* _Giovanni Resta_, Sep 16 2019 *)

%t (* Fast program memoizing the number of instances of a given integer length: *)

%t c[_] = j = 0; c[0] = 1; {0}~Join~Reap[Do[(c[#]++; Set[k, #*10^IntegerLength[c[#]] + c[#]]) &[Boole[j == 0] + IntegerLength[j]]; Sow[k]; j = k, {i, 10^4}]][[-1, -1]] (* _Michael De Vlieger_, Dec 23 2021 *)

%o (MATLAB)

%o length_of_sequence=2000;

%o sequence=zeros(1,length_of_sequence); %allocation for the sequence

%o sequence(1)=11; %start from a(2)=11

%o for i=1:1:(length_of_sequence-1)

%o first=numel(num2str(sequence(1,i))); %defining the first string

%o second=0;

%o for index = 1:size(sequence, 2)

%o     if numel(num2str(sequence(index)))==first

%o     second=second+1; %defining the second string

%o     end

%o end

%o s = strcat(num2str(first),num2str(second));%building a(n+1)

%o sequence(i+1)=str2double(s); %placing the value into a(n+1)

%o end

%o result_sequence=transpose(sequence);

%o (PARI) lista(nn) = {my(va = vector(nn)); va[1] = 0; for (n=2, nn, my(nba = #Str(va[n-1])); my(vn = vector(n-1, k, va[k])); my(nbb = #select(x->(#Str(x)==nba), vn)); va[n] = eval(concat(Str(nba), Str(nbb)));); va;} \\ _Michel Marcus_, Sep 16 2019

%o (Python)

%o from itertools import islice

%o from collections import Counter

%o def agen(): # generator of terms

%o     an, c = 0, Counter()

%o     while True:

%o         yield an

%o         d = len(str(an))

%o         c[d] += 1

%o         an = int(str(d) + str(c[d]))

%o print(list(islice(agen(), 60))) # _Michael S. Branicky_, Nov 12 2022

%K nonn,base

%O 1,2

%A _Bence Bernáth_, Sep 14 2019