A316973 -id:A316973 - 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

A316905 a(n) is the index of the first occurrence of n in A316774.

Original entry on oeis.org

0, 1, 2, 5, 4, 8, 11, 22, 14, 32, 28, 42, 48, 45, 68, 71, 77, 89, 108, 115, 92, 140, 95, 149, 216, 268, 194, 260, 310, 254, 263, 340, 362, 257, 295, 277, 298, 476, 346, 431, 365, 560, 539, 424, 486, 462, 576, 479, 579, 692, 657, 707, 754, 794, 757, 797, 928

Offset: 0

Alois P. Heinz, Jul 18 2018

			a(4) = 4 because A316774(j) = 4 for j in {4,7,12,13,36,49,55} with minimal element 4.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A316774, A316973, A316984, A330440 (a sorted version of this), A330447, A330448.

b:= proc() 0 end:
g:= proc(n) option remember; local t;
      t:= `if`(n<2, n, b(g(n-1))+b(g(n-2)));
      b(t):= b(t)+1; t
    end:
a:= proc() local t, a; t, a:= -1, proc() -1 end;
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1; h:= g(t);
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);

b[_] = 0;
g[n_] := g[n] = Module[{t}, t = If[n < 2, n, b[g[n - 1]] + b[g[n - 2]]];       b[t] = b[t] + 1; t];
a[n_] := Module[{t = -1, a}, a[_] = -1; Module[{h}, While[a[n] == -1, t = t + 1; h = g[t]; If[a[h] == -1, a[h] = t]]; a[n]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 28 2023, after Alois P. Heinz *)

a(n) = min { j >= 0 : A316774(j) = n }.

A316984 a(n) is the index of the last occurrence of n in A316774.

Original entry on oeis.org

0, 1, 6, 16, 55, 488, 314, 367, 279, 562, 519, 799, 500, 694, 792, 876, 1322, 1598, 5826, 1814, 6102, 5345, 2042, 5105, 2949, 3288, 5260, 4971, 6353, 5172, 8596, 3193, 6528, 8065, 8194, 6538, 6881, 6966, 7963, 5993, 8114, 9090, 8457, 8948, 12703, 9994, 6568

Offset: 0

Alois P. Heinz, Jul 18 2018

[It may be that these values are only conjectural. If so, the b-file should be changed to an a-file and marked as conjectured. - N. J. A. Sloane, Dec 14 2019]

			a(4) = 55 because A316774(j) = 4 for j in {4,7,12,13,36,49,55} with maximal element 55.

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A316774, A316905, A316973.

a(n) = max { j >= 0 : A316774(j) = n }.

A337063 a(n) = 1 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)].

Original entry on oeis.org

1, 1, 4, 2, 1, 3, 3, 4, 4, 9, 3, 3, 16, 4, 4, 25, 5, 1, 4, 24, 6, 1, 5, 10, 2, 2, 9, 6, 4, 14, 7, 1, 6, 18, 3, 5, 15, 3, 6, 24, 8, 2, 4, 32, 8, 2, 10, 10, 9, 9, 16, 8, 6, 15, 10, 8, 16, 12, 3, 7, 14, 4, 18, 18, 9, 15, 15, 16, 16, 25, 10, 10, 36, 6, 6, 49, 7, 3

Offset: 0

Alex Lauber, Aug 13 2020

nonn

Does this sequence contain every number?
Does each number appear only a finite number of times?
Starting with a(0)=0 and a(1)=1 gives the same sequence offset by one place.

			a(2) = occurrences of a(1)=1 in [a(0), a(1)]=[1, 1] * occurrences of a(0)=1 in [a(0), a(1)]=[1, 1] = 2*2 = 4.
a(3) = occurrences of a(2)=4 in [a(0), a(1), a(2)]=[1, 1, 4] * occurrences of a(1)=1 in [a(0), a(1), a(2)]=[1, 1, 4] = 1*2 = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..25000
Rémy Sigrist, PARI program for A337063

Cf. A337064 (index of first occurrence of n).

Cf. A316774 (which adds the two previous terms), A316973.

PARI
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See Links section.
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More terms from Rémy Sigrist, Sep 18 2020

A337064 a(n) is the index of the first occurrence of n in A337063, or -1 if n never appears.

Original entry on oeis.org

0, 3, 5, 2, 16, 20, 30, 40, 9, 23, 90, 57, 110, 29, 36, 12, 220, 33, 342, 230, 163, 179, 494, 19, 15, 109, 128, 88, 694, 82, 744, 43, 125, 219, 169, 72, 1060, 373, 253, 85, 1205, 113, 1346, 151, 207, 564, 1726, 131, 75, 332

Offset: 1

Alex Lauber, Aug 13 2020

nonn

419 is the lowest number that does not appear in the first 100000 terms of A337063.

			A337063 starts {1, 1, 4, 2, ...} so a(2) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A337064

Cf. A337063, A316774 (adds the two previous terms), A316973 (similar index for the addition sequence).

PARI
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See Links section.
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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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A316905 a(n) is the index of the first occurrence of n in A316774.

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A316984 a(n) is the index of the last occurrence of n in A316774.

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A337063 a(n) = 1 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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PARI

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A337064 a(n) is the index of the first occurrence of n in A337063, or -1 if n never appears.

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Comments

Examples

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PARI