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

Author

Rémy Sigrist, Nov 26 2019

Keywords

• sign
• look

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).

Examples

			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
		

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.

Crossrefs

o(n) is A330334.

See A329981 for similar sequences.

Programs

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 *)

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])", "))