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.

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, 3, 7, 2, 8, 1, 15, 1, 16, 7, 3, 8, 2, 9, 1, 17, 1, 18, 8, 3, 9, 2, 10, 1, 19, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 3, 10

Offset: 1

Author

David James Sycamore, Aug 05 2023

Keywords

• nonn

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.

Examples

			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,
		

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.

Crossrefs

Cf. A000027, A346175.

Programs

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 *)

Extensions

More terms from Giorgos Kalogeropoulos, Aug 07 2023