A037988 -id:A037988 - cp's OEIS frontend

A272729 a(n) is the number of repetitions of 2n-1 in A272727.

1, 2, 1, 3, 2, 1, 1, 4, 1, 3, 2, 2, 1, 1, 1, 5, 2, 1, 1, 4, 1, 3, 1, 3, 2, 2, 2, 1, 1, 1, 1, 6, 1, 3, 2, 2, 1, 1, 1, 5, 2, 1, 1, 4, 2, 1, 1, 4, 1, 3, 1, 3, 1, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 7, 2, 1, 1, 4, 1, 3, 1, 3, 2, 2, 2, 1, 1, 1, 1, 6, 1, 3, 2, 2, 1, 1, 1, 5, 1, 3, 2, 2, 1, 1, 1, 5

Offset: 1

Ivan Neretin, May 05 2016

nonn

Also, value of A272728 at the n-th local maximum.
Also, the trajectory of 1 under the morphism n->[1,1..1,n+1] (the number of 1's is n-1).
Average value tends to 2.
Number n makes its first appearance at the position 2^(n-1) and has frequency 1/2^n.
Conjectured first differences of A037988 (true for at least 8192 terms). - Sean A. Irvine, Jun 26 2022

			The morphism acts as follows:
1->2; 2->1,3; 3->1,1,4; 4->1,1,1,5; etc.
The trajectory starts as:
1 ->
2 ->
1,3 ->
2,1,1,4 ->
1,3,2,2,1,1,1,5 -> ...
The result of k iterations is a series with 2^(k-1) terms; their sum is 2^k.
If A001511 is laid out in a similar irregular triangle, each row
would contain the same terms, albeit in a different order:
1,
2,
1,3,
1,2,1,4,
1,2,1,3,1,2,1,5...

Cf. A001511, A272727, A272728.

Flatten@NestList[Flatten[Append[ConstantArray[1, # - 1], # + 1] & /@ #] &, {1}, 7]

A094591 a(0) = 1; a(n) = n + (largest element of {a} <= n).

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 11, 12, 16, 17, 20, 22, 24, 25, 26, 27, 32, 34, 35, 36, 40, 41, 44, 45, 48, 50, 52, 54, 55, 56, 57, 58, 64, 65, 68, 70, 72, 73, 74, 75, 80, 82, 83, 84, 88, 90, 91, 92, 96, 97, 100, 101, 104, 105, 108, 110, 112, 114, 116, 117, 118, 119, 120, 121, 128, 130

Offset: 0

Leroy Quet, Jun 07 2004

A self-describing sequence. Pick any number n; this n says: "There are n terms in the sequence which are < 2n". This sequence is the slowest increasing one with this property. See comments on A037988. - Eric Angelini, Jun 15 2007

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Jon Maiga, Computer-generated formulas for A094591, Sequence Machine.

Cf. A037988, A272729.

Block[{a = {1}}, Do[AppendTo[a, i + Last@ TakeWhile[a, # <= i &]], {i, 65}]; a] (* Michael De Vlieger, Sep 04 2017 *)

def aupton(nn):
    alst = [1]
    for n in range(1, nn+1):
        alst.append(n + max(k for k in alst if k <= n))
    return alst
print(aupton(65)) # Michael S. Branicky, Oct 28 2021

From Andrey Zabolotskiy, Oct 28 2021: (Start)
a(n) = A037988(n-1) + 1. [Conjectured by the original author, apparently proved by Eric Angelini.]
The first differences are A272729. [discovered by Sequence Machine] (End)

More terms from Vladeta Jovovic, Jun 11 2004

A359807 a(1) = 0; thereafter a(n) is the largest a(i) + i which is < n among i = 1..n-1.

Original entry on oeis.org

0, 1, 1, 3, 4, 4, 4, 7, 7, 9, 10, 11, 11, 11, 11, 15, 16, 16, 16, 19, 19, 21, 21, 23, 24, 25, 26, 26, 26, 26, 26, 31, 31, 33, 34, 35, 35, 35, 35, 39, 40, 40, 40, 43, 44, 44, 44, 47, 47, 49, 49, 51, 51, 53, 54, 55, 56, 57, 57, 57, 57, 57, 57, 63, 64, 64, 64, 67, 67, 69, 69, 71, 72, 73, 74, 74, 74, 74, 74

Offset: 1

Neal Gersh Tolunsky, Jan 13 2023

nonn

Critical values of Conway's game of one-dimensional phutball (A037988), except for the initial zero.
Conjectured run lengths are A272729, and the terms which occur here are partial sums of A272729.
The next distinct term here occurs at index a(i)+i+1 for every index i.

			For n=3, we see that for i=1 and 2, a(i)+i = 1 and 3, of which only 1 is < n=3, so that a(3)=1.
For n=5, i=1..4 have a(i)+i = 1,3,4,7 and the largest < n=5 is 4 so that a(5)=4.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Cf. A367039, A037988, A272729, A272727, A094591, A367026.

lista(nn) = my(va = vector(nn)); va[1] = 0; for (n=2, nn, va[n] = vecmax(select(x->(xMichel Marcus, Jan 31 2023

{ my (v = 0, m = 0); for (n = 1, 79, if (bittest(m, n-1), v = n-1); print1 (v", "); m = bitor(m, 2^(v+n))) } \\ Rémy Sigrist, Feb 08 2023

A130011 A self-describing sequence. Pick any integer n in the sequence; this n says: "There are n terms in the sequence that are <= 3n". This sequence is the slowest increasing one with this property.

Original entry on oeis.org

1, 4, 5, 12, 15, 16, 17, 18, 19, 20, 21, 36, 37, 38, 45, 48, 51, 54, 57, 60, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Eric Angelini, Jun 15 2007

See comments in A094591 and A037988.

It is not clear in what sense "slowest increasing" is meant in the description of this sequence. The definition requires that there be exactly a(k) terms <= 3 a(k), for any index k. Therefore, a(n+1) > 3n for all indices n of the form n = a(k). Thus, any such sequence has an infinite number of terms a(k) >= 3k-2. The lexicographically first variant A260107, which starts (1, 4, 5, 6, 13, 16, 19, 20, 21, 22, ...), also has all its terms a(k) <= 3k-2, so it cannot be said to increase faster. - M. F. Hasler, Jul 16 2015

Table of n, a(n) for n = 1..27

A260107 Lexicographically first increasing sequence of positive integers such that there are exactly a(k) terms less than or equal to 3*a(k), for each k.

Original entry on oeis.org

1, 4, 5, 6, 13, 16, 19, 20, 21, 22, 23, 24, 25, 40, 41, 42, 49, 50, 51, 58, 61, 64, 67, 70, 73, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 121, 124, 127, 128, 129, 130, 131, 132, 133, 148, 151, 154, 155, 156, 157, 158, 159, 160, 175, 176

Offset: 1

M. F. Hasler, Jul 16 2015

See A130011 and A260139 for other variants of this self-describing sequence.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A130011, A037988, A094591.

l:=[1, 4]:for n from 2 to 20 do for j from l[n-1]+1 to `if`(n=2, l[n]-1, l[n]) do l:=[op(l), max(3*l[n-1], op(l))+1]: od: od: l; # Nathaniel Johnston, Apr 27 2011

a=vector(100,i,1);i=v=1;for(k=2,#a,if(k>a[i],v=3*a[i];i++);a[k]=v++)

a(n) <= 3n-2, and there are infinitely many indices (namely, all those of the form n = a(k)+1 for some k) for which equality holds.

A260139 For any term a(k), there are exactly a(k) terms strictly smaller than 3*a(k); this is the lexicographically first increasing sequence of nonnegative integers with this property.

Original entry on oeis.org

0, 2, 6, 7, 8, 9, 18, 21, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 54, 55, 56, 63, 64, 65, 72, 73, 74, 81, 84, 87, 90, 93, 96, 99, 102, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 162, 165, 168, 169, 170, 171, 172

Offset: 0

M. F. Hasler, Jul 16 2015

Suggested by Eric Angelini, cf. link to SeqFan post.

This sequence has a nice self-similar graph.

			The first term says that there are a(0) = 0 terms < 0.
Then it is not possible to go on with 1, since {0, 1} would be 2 terms < 3*1 = 3.
Thus we must have a(1) = 2 terms < 3*2 = 6; and since we already have {0, 2}, the next must be at least 6.
Therefore, a(2) = 6 is the number of terms < 3*6 = 18, so there must be 3 more:
We have a(3) = 7 terms < 21, a(4) = 8 terms < 24, a(5) = 9 terms < 27.
Now, in view of a(2), the sequence goes on with a(6) = 18 terms < 3*18. This was the 7th term, in view of a(3) the next must be >= 21:
We have a(7) = 21 terms <= 3*21, a(8) = 24 terms <= 3*24, a(9) = 27 terms <= 3*27. Then we can increase by 1 up to index 18:
a(10) = 28 terms <= 3*28, ..., a(17) = 35 terms <= 3*35. This was the 18th term, in view of a(6) the following terms must be >= 3*18 = 54 =: a(18).

M. F. Hasler, Table of n, a(n) for n = 0..999
E. Angelini, Re: A130011 and the definition of "slowest increasing"., SeqFan list, July 13, 2015

Cf. A260107, A130011 and references therein; A037988, A094591 (analogs with 2n instead of 3n).

a=vector(100);a[i=2]=2;for(k=3,#a,a[k]=if(k>a[i],3*a[i++-1],a[k-1]+1))

a(n) <= 3n, with equality for indices of the form n = a(k) for some k.

cp's OEIS Frontend

A272729 a(n) is the number of repetitions of 2n-1 in A272727.

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A094591 a(0) = 1; a(n) = n + (largest element of {a} <= n).

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A359807 a(1) = 0; thereafter a(n) is the largest a(i) + i which is < n among i = 1..n-1.

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A130011 A self-describing sequence. Pick any integer n in the sequence; this n says: "There are n terms in the sequence that are <= 3n". This sequence is the slowest increasing one with this property.

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A260107 Lexicographically first increasing sequence of positive integers such that there are exactly a(k) terms less than or equal to 3*a(k), for each k.

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A260139 For any term a(k), there are exactly a(k) terms strictly smaller than 3*a(k); this is the lexicographically first increasing sequence of nonnegative integers with this property.

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