A309596 If a(n) is not a term of a(0..n-1), then a(n+1) = a(n) - a(n - a(n)); otherwise a(n+1) is the number of terms equal to a(n) in a(0..n-1). Start with a(0)=0, a(1)=0.

0, 0, 1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 1, 5, 3, 2, 2, 3, 3, 4, 1, 6, 4, 2, 4, 3, 5, 1, 7, 1, 8, 4, 4, 5, 2, 5, 3, 6, 1, 9, 1, 10, 6, 2, 6, 3, 7, 1, 11, 5, 4, 6, 4, 7, 2, 7, 3, 8, 1, 12, 11, 1, 13, 8, 2, 8, 3, 9, 1, 14, 7, 4

Offset

0,6

Comments

In other words, if the last term a(n) has not appeared previously, subtract the term a(n - a(n)) from a(n) to get the next term. Otherwise, count the terms equal to the last term a(n) in a(0..n-1) to get the next term.

This sequence has no repeating pattern because of the rule a(n+1) = a(n) - a(n - a(n)).

A new record in the sequence is always the previous record + 1. Therefore all terms are >= 0.

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Example

a(0)=0 (given).
a(1)=0 (given).
a(2)=1: a(1)=0 is a term of a(0..0), therefore a(2) = Number of terms=0 in a(0..0) = 1.
a(3)=1: a(2)=1 is not a term of a(0..1), therefore: a(3) = a(2) - a(2 - 1) = 1 - 0 = 1.
a(4)=1: a(3)=1 is a term of a(0..2), therefore a(4) = Number of terms=1 in a(0..2) = 1.
a(5)=2: a(4)=1 is a term of a(0..3), therefore a(5) = Number of terms=1 in a(0..3) = 2.
a(6)=1: a(5)=2 is not a term of a(0..4), therefore: a(6) = a(5) - a(5 - 2) = 2 - 1 = 1.

Maple

R:= [0, 0]:
for n from 2 to 100 do
  q:= numboccur(R[-1], R[1..-2]);
  if q = 0 then R:= [op(R), R[-1]-R[-1-R[-1]]] else R:= [op(R), q] fi
od:
R; # Robert Israel, Mar 17 2024

Mathematica

s={0, 0}; Do[s1 = If[(c = Count[s[[1;; -2]], s[[-1]]]) == 0, s[[-1]] - s[[-1 -  s[[-1]]]], c]; AppendTo[s, s1], {100}]; s (* Amiram Eldar, Aug 13 2019 *)

Prog

(Python)
sa, n = [0], 0
print(n, sa[n])
while n < 50:
    i, j = 0, 0
    while i < n:
        if sa[i] == sa[n]:
            j = j+1
        i = i+1
    if j == 0:
        a = sa[n] - sa[n-sa[n]]
    else:
        a = j
    n = n+1
    print(n, a)
    sa = sa+[a] # A.H.M. Smeets, Aug 09 2019

Crossrefs

Cf. A181391, A309035.

Sequence in context: A278104 A141672 A141671 * A335442 A226247 A233742

Adjacent sequences: A309593 A309594 A309595 * A309597 A309598 A309599

Keyword

nonn,look

Author

Marc Morgenegg, Aug 09 2019

Status

approved