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

1,2

Comments

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

Links

Samuel Harkness, Table of n, a(n) for n = 1..10000

Samuel Harkness, Scatterplot of the first 2000000 terms

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A133622, A268696, A329985.

Sequence in context: A332604 A132662 A279782 * A132589 A054843 A277427

Adjacent sequences: A358918 A358919 A358920 * A358922 A358923 A358924

Keyword

nonn,hear,look

Author

Samuel Harkness, Dec 06 2022

Status

approved