OFFSET
1,4
COMMENTS
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.
LINKS
Giorgos Kalogeropoulos, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
EXAMPLE
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.
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.
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, 2, 6, 1;
12, 1;
13, 6, 2, 7, 1;
14, 3, 3, 4, 3, 5, 3, 6,
MATHEMATICA
a[1]=1; a[n_]:=a[n]=If[(s=Count[Array[a, n-1], a[n-1]])==1, a[a[n-1]], s-1];
Array[a, 100] (* Giorgos Kalogeropoulos, Aug 07 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore, Aug 05 2023
EXTENSIONS
More terms from Giorgos Kalogeropoulos, Aug 07 2023
STATUS
approved