A330332 -id:A330332 - cp's OEIS frontend

A316774 a(n) = n for n < 2, a(n) = freq(a(n-1),n) + freq(a(n-2),n) for n >= 2, where freq(i,j) is the number of times i appears in [a(0),a(1),...,a(j-1)].

0, 1, 2, 2, 4, 3, 2, 4, 5, 3, 3, 6, 4, 4, 8, 5, 3, 6, 6, 6, 8, 6, 7, 6, 7, 8, 5, 6, 10, 8, 5, 8, 9, 6, 9, 10, 4, 7, 8, 9, 9, 8, 11, 8, 9, 13, 6, 10, 12, 4, 7, 10, 8, 13, 11, 4, 9, 13, 9, 10, 12, 7, 7, 12, 9, 11, 11, 8, 14, 11, 6, 15, 11, 7, 13, 11, 11, 16, 9, 10

Offset: 0

Peter Illig, Jul 12 2018

In other words, a(n) = (number of times a(n-1) has appeared) plus (number of times a(n-2) has appeared). - N. J. A. Sloane, Dec 13 2019
What is the asymptotic behavior of this sequence?
Does it contain every positive integer at least once?
Does it contain every positive integer at most finitely many times?
Additional comments from Peter Illig's "Puzzles" link below (Start):
Sometimes referred to as "The Devil's Sequence" (by me), due to the early presence of three consecutive 6's (and my inability to understand it). The next time a number occurs three times in a row isn't until a(355677).
If each n does appear only finitely many times, approximately how many times does it appear? (It seems to be close to 2n.)
What are the best possible upper/lower bounds on a(n)?
Let r(k) be the smallest n such that {0,1,2,...,k} is contained in {a(0),...,a(n)}. What is the asymptotic behavior of r(k)? (It seems to be close to k^2/2.)
(End)

			For n=4, a(n-1) = a(n-2) = 2, and 2 appears twice in the first 4 terms. So a(4) = 2 + 2 = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..65536
"Horseshoe_Crab" Reddit User, Properties of a Strange, Rather Meta Sequence. [In case this link breaks, the main point of the discussion is to propose the sequence and suggest other initial values. - _N. J. A. Sloane_, Dec 13 2019]
Peter Illig, Problems. [No date, probably 2018]
Samuel B. Reid, Density plot of one billion terms
Rémy Sigrist, Density plot of the first 10000000 terms

Cf. A001462, A316973 (freq(n)), A316905 (when n appears), A316984 (when n last appears), A330439 (total number of times a(n) has appeared so far).
For records see A330330, A330331.
See A306246 and A329934 for similar sequences with different initial conditions.
A330332 considers the frequencies of the three previous terms.

b:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= `if`(n<2, n, b(a(n-1))+b(a(n-2)));
      b(t):= b(t)+1; t
    end:
seq(a(n), n=0..200);  # Alois P. Heinz, Jul 12 2018

a = prev = {0, 1};
Do[
AppendTo[prev, Count[a, prev[[1]]] + Count[a, prev[[2]]]];
AppendTo[a, prev[[3]]];
prev = prev[[2 ;;]] , {78}]
a (* Peter Illig, Jul 12 2018 *)

from itertools import islice
from collections import Counter
def agen():
    a = [0, 1]; c = Counter(a); yield from a
    while True:
        a = [a[-1], c[a[-1]] + c[a[-2]]]; c[a[-1]] += 1; yield a[-1]
print(list(islice(agen(), 80))) # Michael S. Branicky, Oct 13 2022

Definition clarified by N. J. A. Sloane, Dec 13 2019

A330416 a(n) = (number of times a(n-2) has already appeared) + (number of times a(n-3) has already appeared), starting with a(n) = n for n<3.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 3, 3, 4, 6, 5, 3, 2, 5, 7, 5, 4, 4, 7, 8, 6, 3, 3, 8, 12, 8, 4, 4, 9, 12, 7, 3, 5, 10, 11, 5, 2, 6, 9, 7, 5, 6, 10, 10, 7, 6, 8, 10, 9, 8, 7, 8, 12, 12, 10, 8, 9, 12, 11, 9, 7, 7, 13, 16, 9, 2, 7, 11, 14, 12, 4, 7, 13, 17, 12, 3, 8, 15, 16, 9

Offset: 0

Rémy Sigrist, Dec 14 2019

nonn

This sequence is a variant of A330332.

This sequence appears to diverge much more slowly than A330332.

			a(0) = 0.
a(1) = 1.
a(2) = 2.
For n = 3:
- a(1) = 1 has appeared once, a(0) = 0 has appeared once,
- hence a(3) = 1+1 = 2.
For n = 4:
- a(2) = 2 has appeared twice, a(1) = 1 has appeared once,
- hence a(4) = 2+1 = 3.

Rémy Sigrist, C program for A330416
Rémy Sigrist, Density plot of the first 100000000 terms

Cf. A330332.

C
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See Links section.
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Nest[Append[#, Count[#, #[[-2]] ] + Count[#, #[[-3]] ]] &, {0, 1, 2}, 77] (* Michael De Vlieger, Dec 14 2019 *)

cp's OEIS Frontend

A316774 a(n) = n for n < 2, a(n) = freq(a(n-1),n) + freq(a(n-2),n) for n >= 2, where freq(i,j) is the number of times i appears in [a(0),a(1),...,a(j-1)].

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