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

0,3

Comments

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)

Links

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

Example

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.

Maple

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

Mathematica

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 *)

Prog

(Python)
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

Crossrefs

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.

Sequence in context: A161003 A152028 A244318 * A333920 A356164 A246815

Adjacent sequences: A316771 A316772 A316773 * A316775 A316776 A316777

Keyword

nonn,look

Author

Peter Illig, Jul 12 2018

Extensions

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

Status

approved