A358921 a(1) = 1; a(n) is the smallest positive number not among the terms a(n-c .. n-1) where c = the number of times a(n-1) has occurred.

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

Offset: 1

Author

Samuel Harkness, Dec 06 2022

Keywords

• nonn
• hear
• look

Comments

A new number other than 1 is always followed by a 1, so a(n) < n/2 for n > 4.

Examples

			For a(6), a(5) = 1 has occurred 3 times, so the smallest positive integer not in {a(5), a(4), a(3)} = {1, 3, 1} is 2, thus a(6) = 2.
Next, for a(7), a(6) = 2 has occurred 2 times, so the smallest positive integer not in {a(6), a(5)} = {2, 1} is 3, thus a(7) = 3.
Then, for a(8), a(7) = 3 has occurred 2 times, so the smallest positive integer not in {a(7), a(6)} = {3, 2} is 1, thus a(8) = 1.
Now, for a(9), a(8) = 1 has occurred 4 times, so the smallest positive integer not in {a(8), a(7), a(6), a(5)} = {1, 3, 2, 1} is 4, thus a(9) = 4.
The first terms, alongside the number of times they have occurred o(n), are:
  n  a(n)  o(n)
  -  ----  ----
  1     1     1
  2     2     1
  3     1     2
  4     3     1
  5     1     3
  6     2     2
  7     3     2
  8     1     4
  9     4     1
  10    1     5
		

Links

• Samuel Harkness, Table of n, a(n) for n = 1..10000
• Samuel Harkness, Scatterplot of the first 2000000 terms

Crossrefs

Cf. A133622, A268696, A329985.

Programs

Mathematica

V = {1} While[Length[V] < 84, b = 1; While[MemberQ[Take[V, -Count[V, Last[V]]], b], b++ ]; AppendTo[V, b]]; Print[V]

PARI

{ a = o = vector(84); v = 1; for (n=1, #a, print1 (a[n]=v", "); v=setminus([1..n+1], Set(a[n-o[a[n]]+++1..n]))[1]) } \\ Rémy Sigrist, Jan 09 2023