A306246 a(1) = 1, a(2) = 2, and for any n > 2, a(n) = o(n-1) + o(n-2) where o(k) is the number of occurrences of a(k) among a(1), ..., a(k).

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

Offset: 1

Author

Rémy Sigrist, Jan 31 2019

Keywords

• nonn

Comments

The sequence o corresponds to the ordinal transform of the sequence a.

This sequence has a taste of Fibonacci, as each pair of consecutive terms gives rise to the next term.

This sequence is unbounded:

- if the sequence was bounded, say a(n) <= M for any n > 0, then some value in the range 1..M, say t, would appear at least M times among the first M^2 terms, and we would have a value > M after the M-th occurrence of t, a contradiction.

This sequence has interesting graphical features (see Links section).

Examples

			The first terms, alongside o, are:
  n   a(n)  o(n)
  --  ----  ----
   1     1     1
   2     2     1
   3     2     2
   4     3     1
   5     3     2
   6     3     3
   7     5     1
   8     4     1
   9     2     3
  10     4     2
  11     5     2
  12     4     3
  13     5     3
  14     6     1
  15     4     4
  16     5     4
  17     8     1
  18     5     5
  19     6     2
  20     7     1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Density plot of the first 10000000 terms

Crossrefs

See A306251 for the ordinal transform of this sequence.

Programs

PARI

o = vector(18); for (n=1, 79, if (n<=2, v=n, v=pp+p); print1 (v ", "); [pp,p]=[p,o[1+v]++])