A356784 -id:A356784 - cp's OEIS frontend

A357491 Distinct values in A356784, in order of appearance.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 21, 19, 22, 24, 26, 20, 23, 25, 28, 27, 29, 30, 31, 32, 33, 34, 38, 35, 39, 42, 45, 36, 40, 43, 48, 46, 50, 52, 54, 37, 41, 44, 49, 56, 47, 51, 57, 53, 58, 55, 60, 59, 61, 62, 63, 64, 65, 66, 71

Offset: 0

Rémy Sigrist, Oct 01 2022

nonn

We use offset 0 so as to have a permutation of the nonnegative integers.

See A357492 for the inverse.

			Table A357492 begins:
   0,
   0,
   0, 1,
   0, 1, 2, 3,
   0, 1, 2, 4, 3, 5, 6, 7,
   0, 1, 2, 4, 8, 3, 5, 9, 6, 10, 7, 12, 11, 13, 14, 15,
   ...
So the present sequence begins:
   0,
   1,
   2, 3,
   4, 5, 6, 7,
   8, 9, 10, 12, 11, 13, 14, 15,
   ...

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Scatterplot of the first 2^21 terms
Rémy Sigrist, C++ program
Index entries for sequences that are permutations of the natural numbers

Cf. A356784, A357492 (inverse).

a(n) <= 2^k iff n <= 2^k for any k >= 0.

a(2^k) = 2^k for any k >= 0.

A342585 Inventory sequence: record the number of zeros thus far in the sequence, then the number of ones thus far, then the number of twos thus far and so on, until a zero is recorded; the inventory then starts again, recording the number of zeros.

Original entry on oeis.org

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

Offset: 1

Joseph Rozhenko, Mar 16 2021

To get started we ask: how many zero terms are there? Since there are no terms in the sequence yet, we record a '0', and having recorded a '0', we begin again: How many zero terms are there? There is now one 0, so we record a '1' and continue. How many 1's are there? There's currently one '1' in the sequence, so we record a '1' and continue. How many 2's are there? There are no 2's yet, so we record a '0', and having recorded a 0, we begin again with the question "how many zero terms are there?" And so on.
a(46) = 0 because no 8's appear before it; but note a higher number, namely 9, has appeared. - Michael S. Branicky, Mar 16 2021
A similar situation occurs at n=124, where 14 has not yet appeared in the sequence, although 15 has appeared.
Reminiscent of Van Eck's sequence A181391. - N. J. A. Sloane, May 02 2021
From Jan Ritsema van Eck, May 02 2021: (Start)
The first 1000 terms seem to grow more or less in saw-tooth fashion with the largest terms (= the number of 0's), as well as the distance between the 0's, both approximately equal to the inverse triangular numbers A003056 (see attached graph #1).
But the picture changes when we go out to 10000 terms. Around the 1700th term, the 1's become more frequent than the 0's and the largest values are consistently somewhat larger than the inverse triangular numbers. Around the 2500th term the 2's become the most frequent number. Also after some 4000 terms, the largest values become much larger than the inverse triangular numbers. See graph #2. (End)
Comment on the colored plot of the first 1000467 terms, from Hans Havermann, May 02 2021: (Start)
If one is drawing a points-joined graph, it will obscure some of the inherent large-number dynamics. To get around that, this plot joins the points with a green line, superimposing the actual points in blue. This plot was created by Mathematica.
Your browser will likely compress the very large image to window size, so click on it to expand.
The points fall into linear features of the various counts of the various integers. The count for each integer changes as we move towards infinity and hence crosses over (changes place with) other counts unpredictably.
I decided to chart (see the blue text) the twenty largest counts at the rightmost spike which runs from the zero at 997010 to the zero at 1000467. These largest values are for the counts of integers 2 to 21 and appear at a(997013) for the 2-count; a(997014) for the 3-count, ..., and a(997032) for the 21-count.
The counts are 15275, 26832, 40162, 48539, 56364, 54372, 53393, 43588, 37288, 27396, 22425, 16735, 13099, 11460, 9466, 8386, 7191, 6478, 5777, and 5208, respectively. In my text they are sorted largest-to-smallest and written "count @ integer-being-counted": 56364 @ 6, 54372 @ 7, 53393 @ 8, 48539 @ 5, ... 5208 @ 21. (End)
A useful view may be gained by plotting the sequence against itself with an offset. Using the "Plot 2" link in the web page footer, enter "A342585" as sequences 1 and 2. Select "Plot Seq2(n+shift) vs Seq1(n)" and "Draw line segments". Start with "1" as the shift. The sequence appears somewhat like a fan, the first 4 or 5 sectors showing clearly, later sectors overlying each other. Larger shift values effectively compress early sectors into the vertical axis, making later sectors more visible. - Peter Munn, May 08 2021
For a version where a row ends not at the first zero, but rather at the last zero, see A347317. - N. J. A. Sloane, Sep 10 2021
For n around 2.5*10^9, the upper envelope of the sequence seems to be growing roughly like n/50, or maybe like O(n/log(n)). - N. J. A. Sloane, Feb 10 2023

			As an irregular triangle this begins:
   0;
   1,  1,  0;
   2,  2,  2,  0;
   3,  2,  4,  1,  1,  0;
   4,  4,  4,  1,  4,  0;
   5,  5,  4,  1,  6,  2,  1,  0;
   6,  7,  5,  1,  6,  3,  3,  1,  0;
   7,  9,  5,  3,  6,  4,  4,  2,  0;
   8,  9,  6,  4,  9,  4,  5,  2,  1,  3,  0;
   9, 10,  7,  5, 10,  6,  6,  3,  1,  4,  2,  0;
  10, 11,  8,  6, 11,  6,  9,  3,  2,  5,  3,  2,  0;
  ...
For row lengths see A347299. - _N. J. A. Sloane_, Aug 27 2021
From _David James Sycamore_, Oct 18 2021: (Start)
a(1) is 0 because the count is reset, and as yet there is no zero term immediately following another term. a(2) = 1 since the count is reset, a(1) = 0 and a(0) precedes it. The count now increments to terms equal to 1.
a(3) = 1 since a(2) = 1 and a(1) precedes it. a(4) = 0 because there is no term equal to 2 which is immediately preceded by another term.
a(5) = 2 since the count is reset, a(1) = a(4) = 0 and a(0), a(3) respectively, precede them. (End)

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Brady Haran and N. J. A. Sloane, "A Number Sequence with Everything" (the Inventory Sequence A342585), Numberphile video, November 2022.
Hans Havermann, Colored plot of 1000467 terms [See Comments for a description of this plot]
Hugo Pfoertner, Listening to the first 100000 terms of A342585, YouTube video from "Talabfahrer".
Hugo Pfoertner, Hear 1 million terms of A342585, YouTube video from "Talabfahrer", alternative audio conversion.
Luc Rousseau, AWK program for A342585
Rémy Sigrist, Scatterplot of the first 10^6 terms
Rémy Sigrist, Scatterplot of the first 10^7 terms
Rémy Sigrist, Scatterplot of the first 10^8 terms
Rémy Sigrist, Table of n, a(n) for n = 1..100000
Rémy Sigrist, PARI program for A342585
Jan Ritsema van Eck, Graph #1: 1000 terms (blue) and inverse triangular numbers A003056 (orange)
Jan Ritsema van Eck, Graph #2: 10000 terms (blue) and inverse triangular numbers A003056 (orange)
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
Index entries for sequences related to the inventory sequence

Records: A347305 and A348782.
Cf. A181391, A343878, A343880, A347299, A347315, A347316, A347317.
Other inventory-type sequences: A030717, A174382, A333867, A358066, A357443, A356784.
A012257 (cf. also A011784) reverses the inventory process.
See A347062, A347738, A355916, A355917, A355918, A357317 for variants.

# See Links section. - Luc Rousseau, May 02 2021

function [val,arr]=invSeq(N) % val = Nth term, arr = whole array up to N
k=0;
arr=zeros(1,N); % pre-allocate array
for i=1:N
    an=sum((k==arr(2:i)));
    arr(i)=an;
    if an == 0
        k = 0;
    else
        k=k+1;
    end
end
val=arr(end);
end % Ben Cha, Nov 11 2022

a:= proc(n) option remember; local t;
      t:= `if`(a(n-1)=0, 0, b(n-1)+1);
      b(n):=t; add(`if`(a(j)=t, 1, 0), j=1..n-1)
    end: b(1), a(1):= 0$2:
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 16 2021

a[n_] := a[n] = Module[{t}, t = If[a[n-1] == 0, 0, b[n-1]+1];
     b[n] = t; Sum[If[a[j] == t, 1, 0], {j, 1, n-1}]];
b[1] = 0; a[1] = 0;
Array[a, 120] (* Jean-François Alcover, May 03 2021, after Alois P. Heinz *)

A342585_vec(N,c=[],i)=vector(N,j, while(#c<=i||#c<=c[i+1], c=concat(c,0)); c[i+=1]+if(c[1+c[i]]++&&!c[i]||j==1,i=0)) \\ M. F. Hasler, Nov 13 2021

PARI
```
\\ See Links section.
```

def calc(required_value_number):
    values_lst = []
    current_count = 0
    new_value = 0
    for i in range(required_value_number):
        new_value = values_lst.count(current_count)
        values_lst.append(new_value)
        if new_value == 0:
            current_count = 0
        else:
            current_count += 1
    return new_value # Written by Gilad Moyal

from collections import Counter
def aupton(terms):
  num, alst, inventory = 0, [0], Counter([0])
  for n in range(2, terms+1):
    c = inventory[num]
    num = 0 if c == 0 else num + 1; alst.append(c); inventory.update([c])
  return alst
print(aupton(84)) # Michael S. Branicky, Jun 12 2021

# Prints the first 10,068 terms
library("dplyr")
options(max.print=11000)
inventory <- data.frame(1, 0)
colnames(inventory) <- c("n", "an")
value_to_count = 0
n = 1
for(x in 1:128) # Increase the 128 for more terms. The number of terms
                # given is on the order of x^1.9 in the region around 128.
  {
  status <- TRUE
  while(status)
    {
    count <- length(which(inventory$an == value_to_count))
    n = n + 1
    inventory <- rbind(inventory, c(n, count))
    status <- isTRUE(count != 0)
    value_to_count = value_to_count + 1
    }
  value_to_count = 0
  }
inventory # Damon Lay, Nov 10 2023

A357443 Inventory sequence, second version: record where the 1's, 2's, etc. are located starting with a(1) = 1, a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 4, 6, 7, 8, 1, 2, 3, 5, 9, 4, 6, 10, 7, 11, 8, 13, 12, 14, 15, 16, 1, 2, 3, 5, 9, 17, 4, 6, 10, 18, 7, 11, 19, 8, 13, 22, 12, 20, 14, 23, 15, 25, 16, 27, 21, 24, 26, 29, 28, 30, 31, 32, 1, 2, 3, 5, 9, 17, 33, 4, 6, 10, 18, 34

Offset: 1

Ctibor O. Zizka, Oct 29 2022, edited by N. J. A. Sloane, Nov 07 2022. (Because of a missing term in the initial submission, the definition could be interpreted in two ways: A358066 was the first interpretation, this is the second.)

Essentially the same as A358066, except that at stage k we record where all the numbers from 1 to 2^k have appeared in the sequence in earlier stages. (In A358066 we only look at where the numbers from 1 to k have appeared.)

This means that when the sequence is displayed as a triangle, the row corresponding to stage k >= 1 is a permutation of the numbers from 1 to 2^k.

			As an irregular triangle, the sequence begins:
1, 1,
1, 2,
1, 2, 3, 4,
1, 2, 3, 5, 4, 6, 7, 8,
1, 2, 3, 5, 9, 4, 6, 10, 7, 11, 8, 13, 12, 14, 15, 16,
1, 2, 3, 5, 9, 17, 4, 6, 10, 18, 7, 11, 19, 8, 13, 22, 12, 20, 14, 23, 15, 25, 16, 27, 21, 24, 26, 29, 28, 30, 31, 32,
...

Cf. A342585, A356784, A358066.

a(n) = A356784(n-1) + 1. - Rémy Sigrist, Nov 08 2022

More terms from Rémy Sigrist, Nov 08 2022

A358066 Inventory sequence: record where the 1's, 2's, etc. are located starting with a(1) = 1, a(2) = 1 (see example).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 4, 6, 7, 1, 2, 3, 5, 9, 4, 6, 10, 7, 11, 8, 13, 1, 2, 3, 5, 9, 16, 4, 6, 10, 17, 7, 11, 18, 8, 13, 21, 12, 19, 1, 2, 3, 5, 9, 16, 28, 4, 6, 10, 17, 29, 7, 11, 18, 30, 8, 13, 21, 34, 12, 19, 31, 14, 22, 35, 1, 2, 3, 5, 9, 16, 28, 46, 4, 6, 10, 17, 29, 47, 7, 11, 18, 30, 48

Offset: 1

Ctibor O. Zizka, Oct 29 2022

			At stage n >= 1 we only look at the numbers 1 up to n, and ignore numbers bigger than n.
Stage 0: start with a(1) = 1, a(2) = 1.
Stage 1: we see 1's at 1,2, so we adjoin 1,2, getting 1,1, 1,2.
Stage 2: we see 1's at 1,2,3, and 2's at 4, so we adjoin 1,2,3,4, getting 1,1,1,2, 1,2,3,4.
Stage 3: we see 1's at 1,2,3,5, 2's at 4,6 and 3's at 7, so we adjoin 1,2,3,5,4,6,7, getting 1,1,1,2,1,2,3,4, 1,2,3,5,4,6,7.
Stage 4: we see 1's at 1,2,3,5,9, 2's at 4,6,10, 3's at 7,11, 4's at 8,13, so we adjoin 1,2, ..., 8,13 and so on.
We obtain an irregular triangle by writing the results of the stages as separate rows:
1, 1,
1, 2,
1, 2, 3, 4,
1, 2, 3, 5, 4, 6, 7,
1, 2, 3, 5, 9, 4, 6, 10, 7, 11, 8, 13,
1, 2, 3, 5, 9, 16, 4, 6, 10, 17, 7, 11, 18, 8, 13, 21, 12, 19,
1, 2, 3, 5, 9, 16, 28, 4, 6, 10, 17, 29, 7, 11, 18, 30, 8, 13, 21, 34, 12, 19, 31, 14, 22, 35,
... (_N. J. A. Sloane_, Nov 07 2022)

See A357443 for another version.

Cf. A030717, A055187, A217780, A342585, A357317, A356784.

terms = [1, 1]
for i in range(1,11):
    new_terms = []
    for j in range(1, i+1):
        for k in range(len(terms)):
            if terms[k] == j: new_terms.append(k+1)
    terms.extend(new_terms)
print(terms) # Gleb Ivanov, Nov 01 2022

A358085 Inventory of positions ordered by binary lengths of terms, as an irregular table; the first row contains 1, subsequent rows contains the 1-based positions of terms with binary length 1, followed by positions of terms with binary length 2, 3, etc. in prior rows flattened.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 4, 6, 7, 8, 1, 2, 3, 5, 9, 4, 6, 7, 10, 11, 8, 12, 13, 14, 15, 16, 1, 2, 3, 5, 9, 17, 4, 6, 7, 10, 11, 18, 19, 8, 12, 13, 14, 15, 20, 22, 23, 24, 16, 21, 25, 26, 27, 28, 29, 30, 31, 32

Offset: 1

Rémy Sigrist, Oct 30 2022

The n-th row contains A011782(n-1) terms, and is a permutation of 1..A011782(n-1).

			Table begins:
    1,
    1,
    1, 2,
    1, 2, 3, 4,
    1, 2, 3, 5, 4, 6, 7, 8,
    1, 2, 3, 5, 9, 4, 6, 7, 10, 11, 8, 12, 13, 14, 15, 16,
    ...
For n = 6:
- the terms in rows 1..5 are: 1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 4, 6, 7, 8,
- terms with binary length 1 are at positions: 1, 2, 3, 5, 9,
- terms with binary length 2 are at positions: 4, 6, 7, 10, 11,
- terms with binary length 3 are at positions: 8, 12, 13, 14, 15,
- terms with binary length 4 are at positions: 16,
- so row 6 is: 1, 2, 3, 5, 9, 4, 6, 7, 10, 11, 8, 12, 13, 14, 15, 16.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program
Rémy Sigrist, Scatterplot of the first 2^20 terms

Cf. A011782, A070939, A342585, A356784, A358121.

PARI
```
See Links section.
```

A358090 Partial inventory of positions as an irregular table; rows 1 and 2 contain 1, for n > 2, row n contains the 1-based positions of 1's, followed by the positions of 2's, 3's, etc. in rows n-2 and n-1 flattened.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 3, 2, 4, 5, 1, 4, 2, 6, 3, 5, 7, 8, 1, 6, 3, 8, 2, 10, 4, 7, 5, 11, 9, 12, 13, 1, 9, 3, 13, 5, 11, 2, 15, 6, 17, 4, 10, 7, 16, 8, 12, 19, 14, 18, 20, 21, 1, 14, 5, 20, 3, 16, 7, 24, 9, 18, 2, 22, 8, 26, 4, 28, 11, 15, 6, 25, 10, 19, 12, 29, 13, 17, 31, 21, 27, 23, 32, 30, 33, 34

Offset: 1

Rémy Sigrist, Oct 30 2022

This sequence is a variant of A356784; here we consider two prior rows, there all prior rows, hence the term "partial" in the name.

The n-th row contains A000045(n) terms, and is a permutation of 1..A000045(n).

			Table begins:
    1,
    1,
    1, 2,
    1, 2, 3,
    1, 3, 2, 4, 5,
    1, 4, 2, 6, 3, 5, 7, 8,
    1, 6, 3, 8, 2, 10, 4, 7, 5, 11, 9, 12, 13,
    ...
For n = 7:
- the terms in rows 5 and 6 are: 1, 3, 2, 4, 5, 1, 4, 2, 6, 3, 5, 7, 8,
- positions of 1's are: 1, 6,
- positions of 2's are: 3, 8,
- positions of 3's are: 2, 10,
- positions of 4's are: 4, 7,
- positions of 5's are: 5, 11,
- positions of 6's are: 9,
- positions of 7's are: 12,
- positions of 8's are: 13,
- so row 7 is: 1, 6, 3, 8, 2, 10, 4, 7, 5, 11, 9, 12, 13.

Rémy Sigrist, Table of n, a(n) for n = 1..10945
Rémy Sigrist, Scatterplot of the first 832039 terms
Rémy Sigrist, PARI program

See A358120 for a similar sequence.

Cf. A000045, A001611, A342585, A356784, A358123.

PARI
```
See Links section.
```

T(n, 1) = 1.

T(n, 2) = A001611(n-2) for n > 2.

A358120 Partial inventory of positions as an irregular table; rows 1 and 2 contain 1, for n > 2, row n contains the 1-based positions of 1's, followed by the positions of 2's, 3's, etc. in rows n-1 and n-2 flattened.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 6, 5, 8, 3, 7, 2, 4, 1, 9, 7, 13, 5, 11, 8, 10, 3, 12, 2, 6, 4, 1, 14, 11, 20, 9, 18, 13, 21, 5, 16, 12, 15, 3, 19, 7, 17, 2, 8, 6, 10, 4, 1, 22, 17, 32, 13, 30, 21, 34, 9, 26, 19, 33, 15, 24, 18, 28, 5, 23, 20, 29, 3, 27, 11, 31, 7, 25, 2, 12, 10, 16, 6, 14, 4, 8

Offset: 1

Rémy Sigrist, Oct 30 2022

The n-th row contains A000045(n) terms, and is a permutation of 1..A000045(n).

			Table begins:
    1,
    1,
    1, 2,
    1, 3, 2,
    1, 4, 3, 5, 2,
    1, 6, 5, 8, 3, 7, 2, 4,
    1, 9, 7, 13, 5, 11, 8, 10, 3, 12, 2, 6, 4,
    ...
For n = 7:
- terms in rows 6 and 5 are: 1, 6, 5, 8, 3, 7, 2, 4, 1, 4, 3, 5, 2,
- positions of 1's are: 1, 9,
- positions of 2's are: 7, 13,
- positions of 3's are: 5, 11,
- positions of 4's are: 8, 10,
- positions of 5's are: 3, 12,
- positions of 6's are: 2,
- positions of 7's are: 6,
- positions of 8's are: 4,
- so row 7 is: 1, 9, 7, 13, 5, 11, 8, 10, 3, 12, 2, 6, 4.

Rémy Sigrist, Table of n, a(n) for n = 1..10945
Rémy Sigrist, PARI program
Rémy Sigrist, Scatterplot of the first 832039 terms

See A358090 for a similar sequence.

Cf. A000045, A001611, A342585, A356784, A358124.

PARI
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See Links section.
```

T(n, 1) = 1.

T(n, 2) = A001611(n-1) for n > 2.

A357492 Inverse permutation to A357491.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 20, 24, 19, 21, 25, 22, 26, 23, 28, 27, 29, 30, 31, 32, 33, 34, 36, 40, 48, 35, 37, 41, 49, 38, 42, 50, 39, 44, 53, 43, 51, 45, 54, 46, 56, 47, 58, 52, 55, 57, 60, 59, 61, 62, 63, 64, 65, 66, 68

Offset: 0

Rémy Sigrist, Oct 01 2022

nonn

			A357491(89) = 100, hence a(100) = 89.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, C++ program
Index entries for sequences that are permutations of the natural numbers

Cf. A356784, A357491.

A358086 Inventory of positions ordered by odd parts of terms, as an irregular table; the first row contains 1, subsequent rows contains the 1-based positions of terms with odd part 1, followed by positions of terms with odd part 3, 5, etc. in prior rows flattened.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 8, 7, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 7, 11, 14, 13, 16, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 17, 18, 20, 23, 32, 7, 11, 14, 19, 22, 26, 13, 21, 25, 16, 28, 30, 24, 29, 31, 27

Offset: 1

Rémy Sigrist, Oct 30 2022

The n-th row contains A011782(n-1) terms, and is a permutation of 1..A011782(n-1).

			Table begins:
    1,
    1,
    1, 2,
    1, 2, 3, 4,
    1, 2, 3, 4, 5, 6, 8, 7,
    1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 7, 11, 14, 13, 16,
    ...
For n = 6:
the terms in rows 1..5 are: 1, 1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 8, 7,
- terms with odd part 1 are at positions: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15,
- terms with odd part 3 are at positions: 7, 11, 14,
- terms with odd part 5 are at positions: 13,
- terms with odd part 7 are at positions: 16,
- so row 6 is: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 7, 11, 14, 13, 16.

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Rémy Sigrist, PARI program
Rémy Sigrist, Scatterplot of the first 2^20 terms

Cf. A000265, A011782, A342585, A356784, A358085, A358122.

PARI
```
See Links section.
```

A358503 Positions inventory sequence: for stage k >= 2 we record where all the numbers from the two previous stages have appeared, starting with a(0) = 0, a(1) = 0.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 2, 3, 4, 5, 6, 4, 5, 7, 6, 8, 9, 10, 11, 7, 8, 9, 12, 10, 13, 11, 15, 14, 16, 17, 18, 19, 12, 13, 15, 14, 20, 16, 21, 17, 22, 18, 24, 19, 26, 23, 25, 28, 27, 29, 30, 31, 32, 20, 21, 22, 24, 26, 23, 33, 25, 34, 28, 36, 27, 35, 29, 38, 30, 40, 31, 42

Offset: 0

Ctibor O. Zizka, Nov 21 2022

When the sequence is displayed as a triangle, the row corresponding to stage k >= 2 is a permutation of the numbers from Fibonacci(k) - 1 to Fibonacci(k+2) - 2.

			As an irregular triangle, the sequence begins:
  0;
  0;
  0,  1;
  1,  2,  3;
  2,  3,  4,  5,  6;
  4,  5,  7,  6,  8,  9, 10, 11;
  7,  8,  9, 12, 10, 13, 11, 15, 14, 16, 17, 18, 19;
  ...
At stage 5 we look at two previous stages 3 and 4 and see the
positions of 1's: 4;
positions of 2's: 5, 7;
positions of 3's: 6, 8;
positions of 4's: 9;
positions of 5's: 10;
positions of 6's: 11;
thus stage 5 is 4, 5, 7, 6, 8, 9, 10, 11.

Cf. A000045, A000071, A001911, A356784, A357317, A357443.

cp's OEIS Frontend

A357491 Distinct values in A356784, in order of appearance.

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A342585 Inventory sequence: record the number of zeros thus far in the sequence, then the number of ones thus far, then the number of twos thus far and so on, until a zero is recorded; the inventory then starts again, recording the number of zeros.

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A357443 Inventory sequence, second version: record where the 1's, 2's, etc. are located starting with a(1) = 1, a(2) = 1.

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A358066 Inventory sequence: record where the 1's, 2's, etc. are located starting with a(1) = 1, a(2) = 1 (see example).

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A358085 Inventory of positions ordered by binary lengths of terms, as an irregular table; the first row contains 1, subsequent rows contains the 1-based positions of terms with binary length 1, followed by positions of terms with binary length 2, 3, etc. in prior rows flattened.

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A358090 Partial inventory of positions as an irregular table; rows 1 and 2 contain 1, for n > 2, row n contains the 1-based positions of 1's, followed by the positions of 2's, 3's, etc. in rows n-2 and n-1 flattened.

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A358120 Partial inventory of positions as an irregular table; rows 1 and 2 contain 1, for n > 2, row n contains the 1-based positions of 1's, followed by the positions of 2's, 3's, etc. in rows n-1 and n-2 flattened.

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A357492 Inverse permutation to A357491.

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