A329985 a(1) = 1 and for n > 0, a(n+1) = a(k) - a(n) where k is the number of terms equal to a(n) among the first n terms.

1, 0, 1, -1, 2, -1, 1, 0, 0, 1, -2, 3, -2, 2, -2, 3, -3, 4, -3, 3, -2, 1, 1, -2, 4, -4, 5, -4, 4, -3, 4, -5, 6, -5, 5, -5, 6, -6, 7, -6, 6, -5, 4, -2, 1, 0, -1, 2, -1, 0, 2, -3, 2, 0, -1, 3, -4, 5, -4, 3, -1, 0, 1, -1, 2, -3, 5, -6, 7, -7, 8, -7, 7, -6, 5, -3

Offset

1,5

Comments

In other words, for n > 0, a(n+1) = a(o(n)) - a(n) where o is the ordinal transform of the sequence.

The sequence has interesting graphical features (see plot in Links section).

Links

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Rémy Sigrist, Density plot of the first 2^22 terms

N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

Example

The first terms, alongside their ordinal transform, are:
  n   a(n)  o(n)
  --  ----  ----
   1     1     1
   2     0     1
   3     1     2
   4    -1     1
   5     2     1
   6    -1     2
   7     1     3
   8     0     2
   9     0     3
  10     1     4

Mathematica

A={1}; For[n=2, n<=76, n++, A=Append[A, Part[A, Count[Table[Part[A, i], {i, 1, n-1}], Part[A, n-1]]]-Part[A, n-1]]]; A (* Joshua Oliver, Nov 26 2019 *)
Nest[Append[#, #[[Count[#, #[[-1]] ] ]] - #[[-1]]] &, {1}, 75] (* Michael De Vlieger, Dec 01 2019 *)

Prog

(PARI) for (n=1, #(a=vector(76)), print1 (a[n]=if (n==1, 1, a[sum(k=1, n-1, a[k]==a[n-1])]-a[n-1])", "))

Crossrefs

o(n) is A330334.

See A329981 for similar sequences.

Sequence in context: A277349 A078807 A208249 * A029422 A351356 A152800

Adjacent sequences: A329982 A329983 A329984 * A329986 A329987 A329988

Keyword

sign,look

Author

Rémy Sigrist, Nov 26 2019

Status

approved