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A364749 a(1) = 1. Thereafter, if a(n-1) is a novel term a(n) = a(a(n-1)), otherwise a(n) is the number of times a(n-1) has been repeated.

%I A364749 #16 Aug 08 2023 01:43:04
%S A364749 1,1,1,2,1,3,1,4,2,1,5,1,6,3,1,7,1,8,4,1,9,2,2,3,2,4,2,5,1,10,1,11,5,
%T A364749 2,6,1,12,1,13,6,2,7,1,14,3,3,4,3,5,3,6,3,7,2,8,1,15,1,16,7,3,8,2,9,1,
%U A364749 17,1,18,8,3,9,2,10,1,19,4,4,5,4,6,4,7,4,8,4,9,3,10
%N A364749 a(1) = 1. Thereafter, if a(n-1) is a novel term a(n) = a(a(n-1)), otherwise a(n) is the number of times a(n-1) has been repeated.
%C A364749 Based on A346175, except that this sequence has offset 1, and begins a(1) = 1. When a(n-1) is a repeated term, seen k times up to and including itself, a(n) = k-1, the number of repeats of a(n-1). A record term a(m) = r beyond a(2) arises consequent to a(m-1) = 1, and is the number of times 1 has been repeated so far. The subsequence {a(r)} recovers the original sequence, which is fractal. The records subsequence is A000027.
%H A364749 Giorgos Kalogeropoulos, <a href="/A364749/b364749.txt">Table of n, a(n) for n = 1..10000</a>
%H A364749 Michael De Vlieger, <a href="/A364749/a364749.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%e A364749 a(1) = 1 is a novel term, seen for the first time, so a(2) = a(a(1)) = a(1) = 1. 1 has now been repeated once so a(3) = 1. Now 1 has been repeated twice, so a(4) = 2, a novel term, meaning that a(5) = a(a(4)) = a(2) = 1.
%e A364749 The sequence can be represented as an irregular table wherein row n starts with the n-th record term and ends with a 1 prior to the next record term, which starts the next row. The first column of the table is the records subsequence, A000027, and the second column is a copy the sequence itself.
%e A364749    1,  1, 1;
%e A364749    2,  1;
%e A364749    3,  1;
%e A364749    4,  2, 1;
%e A364749    5,  1;
%e A364749    6,  3, 1;
%e A364749    7,  1;
%e A364749    8,  4, 1;
%e A364749    9,  2, 2, 3, 2, 4, 2, 5, 1;
%e A364749    10, 1;
%e A364749    11, 5, 2, 6, 1;
%e A364749    12, 1;
%e A364749    13, 6, 2, 7, 1;
%e A364749    14, 3, 3, 4, 3, 5, 3, 6,
%t A364749 a[1]=1;a[n_]:=a[n]=If[(s=Count[Array[a,n-1],a[n-1]])==1,a[a[n-1]],s-1];
%t A364749 Array[a,100] (* _Giorgos Kalogeropoulos_, Aug 07 2023 *)
%Y A364749 Cf. A000027, A346175.
%K A364749 nonn
%O A364749 1,4
%A A364749 _David James Sycamore_, Aug 05 2023
%E A364749 More terms from _Giorgos Kalogeropoulos_, Aug 07 2023