A364784 a(n) = n for n <= 2. Thereafter if a(n-1) is a novel term, a(n) = a(a(k)) where k is the greatest prior term < a(n-1); otherwise, a(n) = number of times a(n-1) has been repeated.

1, 2, 1, 1, 2, 1, 3, 2, 2, 3, 1, 4, 1, 5, 1, 6, 2, 4, 1, 7, 1, 8, 3, 2, 5, 1, 9, 2, 6, 1, 10, 2, 7, 1, 11, 3, 3, 4, 2, 8, 1, 12, 1, 13, 4, 3, 5, 2, 9, 1, 14, 1, 15, 5, 3, 6, 2, 10, 1, 16, 1, 17, 6, 3, 7, 2, 11, 1, 18, 2, 12, 1, 19, 4, 4, 5, 4, 6, 4, 7, 3, 8, 2

Offset: 1

Author

David James Sycamore, Aug 07 2023

Keywords

• nonn
• tabf

Comments

The definition is similar to that of A364749.

With the exception of a(3) = a(4) = 1, every term a(r-1) = 1 occurs prior to record term a(r), and a(r) is the number of times 1 has been repeated so far.

The subsequence of records is A000027, and the subsequence {a(r)} is a copy of the sequence itself, which is fractal (see Example).

Examples

			The given terms are a(1) = 1 and a(2) = 2. Since 2 is a novel term and 1 is the greatest prior term < 2, a(3) = a(1) = 1, and since a(3) is the second occurrence of 1, a(4) = 1 (the number of times 1 has been repeated). Now 1 has occurred 3 times so a(5) = 2, and so on.
The sequence can be represented as an irregular table in which the n-th row starts with the n-th record, and ends with the term = 1 which precedes the next record. Thus the first column is A000027, and the second column is the sequence itself.
The table begins:
  1;
  2,1,1,2,1;
  3,2,2,3,1;
  4,1;
  5,1;
  6,2,4,1;
  7,1;
  8,3,2,5,1;
  9,2,6,1;
  10,2,7,1;
  11,3,3,4,2,8,1;
  12,1;
  13,4,3,5,2,9,1;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Crossrefs

Cf. A000027, A364749.

Programs

Mathematica

nn = 1000; c[] = 0; Array[Set[a[#], #] &, 2]; c[1] = 1; Do[a[n] = If[c[#] == 0, c[#]++; k = # - 1; While[c[k] == 0, k--]; a[k], c[#]; c[#]++] &@ a[n - 1], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Aug 07 2023 *)

Extensions

More terms from Michael De Vlieger, Aug 07 2023