A227736 -id:A227736 - cp's OEIS frontend

A227740 Integers from 0 to A037834(n) followed by integers from 0 to A037834(n+1) and so on.

0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3

Offset: 1

Antti Karttunen, Jul 25 2013

nonn

Equivalently, integers from 0 to A005811(n)-1 followed by integers from 0 to A005811(n+1)-1 and so on.

Vincenzo Librandi, Table of n, a(n) for n = 1..2272

Cf. A227737, A227741, A005811, A037834, A173318.

Cf. also A227736, A227738, A227739.

Table[Range[0, #] &@ Total@ Flatten@ Map[Abs@ Differences@ # &,
Partition[IntegerDigits[n, 2], 2, 1]], {n, 34}] // Flatten (* Michael De Vlieger, May 09 2017 *)

(define (A227740 n) (- n (+ 1 (A173318 (- (A227737 n) 1)))))

a(n) = n - (1 + A173318(A227737(n)-1)).

A163510 Irregular table read by rows: Write n in binary. For each 1, the m-th term of row n is the number of 0's between the m-th 1, reading right to left, and the (m-1)th 1 (or the right side of the number if m-1 = 0).

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 0, 0, 3, 0, 2, 1, 1, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 3, 1, 2, 0, 0, 2, 2, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 3, 0, 0, 2, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 4, 1, 3, 0, 0, 3, 2, 2, 0, 1, 2, 1, 0, 2, 0, 0, 0, 2, 3, 1, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1

Offset: 1

Leroy Quet, Jul 29 2009

Row n contains exactly A000120(n) terms, for each n.
All odd-numbered rows begin with 0. All even-numbered rows begin with a positive integer.
Can be used to compute the permutation A163511.

			Table begins as:
  Row  n in    Terms on
   n   binary  that row
   1      1    0; (the distance of 1-bit from the right edge is zero)
   2     10    1; (the distance of 1-bit from the right edge is one)
   3     11    0,0;
   4    100    2;
   5    101    0,1; (the least significant 1-bit is zero steps away from the right edge, and there is one zero between those two 1-bits)
   6    110    1,0;
   7    111    0,0,0;
   8   1000    3;
   9   1001    0,2;
  10   1010    1,1;
  11   1011    0,0,1;
  12   1100    2,0;
  13   1101    0,1,0;
  14   1110    1,0,0;
  15   1111    0,0,0,0;
  16  10000    4;

Antti Karttunen, Table of n, a(n) for n = 1..11265

Cf. A000120, A006068, A100922, A227186, A243067, A163511, A227736.

Equals A228351-1, termwise.

Table[Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]], {n, 46}] // Flatten (* Michael De Vlieger, Jul 25 2016 *)

from itertools import count, islice
def A163510_gen(): # generator of terms
    for n in count(1):
        k = n
        while k:
            yield (s:=(~k&k-1).bit_length())
            k >>= s+1
A163510_list = list(islice(A163510_gen(),30)) # Chai Wah Wu, Jul 17 2023

(define (A163510 n) (- (A227186bi (A006068 (A100922 (- n 1))) (A243067 n)) 1))
;; See A227186 for A227186bi. - Antti Karttunen, Jun 19 2014

a(n) = A227186(A006068(A100922(n-1)), A243067(n)) - 1. - Antti Karttunen, Jun 19 2014

Additional terms computed and Example section added by Antti Karttunen, Jun 19 2014

A227741 Simple self-inverse permutation of natural numbers: List each block of A005811(n) numbers from A173318(n-1)+1 to A173318(n) in reverse order.

Original entry on oeis.org

1, 3, 2, 4, 6, 5, 9, 8, 7, 11, 10, 12, 14, 13, 17, 16, 15, 21, 20, 19, 18, 24, 23, 22, 26, 25, 29, 28, 27, 31, 30, 32, 34, 33, 37, 36, 35, 41, 40, 39, 38, 44, 43, 42, 48, 47, 46, 45, 53, 52, 51, 50, 49, 57, 56, 55, 54, 60, 59, 58, 62, 61, 65, 64, 63, 69, 68, 67, 66

Offset: 1

Antti Karttunen, Jul 25 2013

nonn

This permutation maps between such irregular tables as A101211 and A227736 which are otherwise identical, except for the order in which the lengths of runs have been listed. In other words, A227736(n) = A101211(a(n)) and vice versa, A101211(n) = A227736(a(n)).

Antti Karttunen, Table of n, a(n) for n = 1..1001
Index entries for sequences that are permutations of the natural numbers

Cf. A227742 (gives the fixed points).

Cf. also A227737, A227740, A005811, A173318.

(define (A227741 n) (- (A173318 (A227737 n)) (A227740 n)))

a(n) = A173318(A227737(n)) - A227740(n).

A318927 Take the binary expansion of n, starting with the most significant bit, and concatenate the lengths of the runs.

Original entry on oeis.org

1, 11, 2, 12, 111, 21, 3, 13, 121, 1111, 112, 22, 211, 31, 4, 14, 131, 1211, 122, 1112, 11111, 1121, 113, 23, 221, 2111, 212, 32, 311, 41, 5, 15, 141, 1311, 132, 1212, 12111, 1221, 123, 1113

Offset: 1

N. J. A. Sloane, Sep 09 2018

Obviously this compressed notation is useful only for n < 1023. A101211 is a version which works for all n.

			n, binary, run lengths, -> a(n)
1, [1], [1] -> 1
2, [1, 0], [1, 1] -> 11
3, [1, 1], [2] -> 2
4, [1, 0, 0], [1, 2] -> 12
5, [1, 0, 1], [1, 1, 1] -> 111
6, [1, 1, 0], [2, 1] -> 21
7, [1, 1, 1], [3] -> 3
...

Rémy Sigrist, Table of n, a(n) for n = 1..1022

Cf. A101211 (without concatenation, as rows), A227736 (rows reversed), A318926 (reverse concatenation).

Cf. A382255 (Heinz numbers instead of concatenation of the run lengths).

Array[FromDigits@ Flatten[IntegerDigits@ Length[#] & /@ Split@ IntegerDigits[#, 2]] &, 40] (* Michael De Vlieger, Feb 17 2022 *)

a(n) = { my(d=[], r); while(n, n>>=r=valuation(n+n%2, 2); d=concat(digits(r), d)); fromdigits(d) } \\ Rémy Sigrist, Feb 17 2022, edited by M. F. Hasler, Mar 11 2025

from itertools import groupby
def A318927(n): return int(''.join(str(len(list(g))) for k, g in groupby(bin(n)[2:]))) # Chai Wah Wu, Mar 11 2022

A318926 Take the binary expansion of n, starting with the least significant bit, and concatenate the lengths of the runs.

Original entry on oeis.org

1, 11, 2, 21, 111, 12, 3, 31, 121, 1111, 211, 22, 112, 13, 4, 41, 131, 1121, 221, 2111, 11111, 1211, 311, 32, 122, 1112, 212, 23, 113, 14, 5, 51, 141, 1131, 231, 2121, 11121, 1221, 321, 3111, 12111, 111111, 21111, 2211, 11211, 1311, 411, 42, 132, 1122, 222, 2112, 11112, 1212, 312, 33, 123

Offset: 1

N. J. A. Sloane, Sep 09 2018

Obviously this compressed notation is useful only for n < 2047. A227736 is a version which works for all n. [Corrected by M. F. Hasler, Mar 12 2025]

			n, binary, run lengths, -> a(n)
1, [1], [1] -> 1
2, [0, 1], [1, 1] -> 11
3, [1, 1], [2] ->  2
4, [0, 0, 1], [2, 1] -> 21
5, [1, 0, 1], [1, 1, 1] -> 111
6, [0, 1, 1], [1, 2] -> 12
7, [1, 1, 1], [3] -> 3
8, [0, 0, 0, 1], [3, 1] ->  31,
...
From _M. F. Hasler_, Mar 12 2025: (Start)
For n = 1023 = 2^10-1, n = '1'*10 in binary, so there is only one run of length 10, whence a(n) = 10. This value cannot occur at any other index n.
For n = 1024 = 2^10, n = '1'+'0'*10 in binary, so the run lengths, from right to left, are [10, 1], whence a(n) = 101. The only other index n for which this value occurs is n = 2^101-1.
For n = 1025 = 2^10+1, n = '1'+'0'*9+'1' in binary, so a(n) = 191. This values occurs for the second time as a(n = 2^19), for the third time for a(n = 2^92-2), and for the 4th and last time as a(n = 2^191-1).
Similarly, a(1026) = 1181 appears for the second time at n = 2^19 + 1 = 524289;
  a(1027) = 281 occurs a 2nd, 3rd and 4th time at n = 2^28, (2^81-1)*2 and 2^281-1.
The first duplicate value occurs as a(2047 = 2^11-1) = 11 = a(2). (End)

Paolo Xausa, Table of n, a(n) for n = 1..10000
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. - _N. J. A. Sloane_, Sep 09 2018. See Procedure 1.

Cf. A227736 (run lengths in rows instead of concatenation), A101211 (rows in reverse order), A318927 (concatenation in reverse order).

A318926[n_] := FromDigits[Flatten[IntegerDigits[Map[Length, Split[Reverse[IntegerDigits[n, 2]]]]]]];
Array[A318926, 100] (* Paolo Xausa, Mar 16 2025 *)

A318926(n)=eval(strjoin(Vecrev(A101211_row(n)))); \\ M. F. Hasler, Mar 11 2025

from itertools import groupby
def A318926(n): return int(''.join(str(len(list(g))) for k, g in groupby(bin(n)[:1:-1]))) # Chai Wah Wu, Mar 11 2022

More terms from M. F. Hasler, Mar 12 2025

A341840 Square array T(n, k), n, k >= 0, read by antidiagonals; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) is the intersection of R(n) and of R(k).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 3, 3, 0, 0, 0, 1, 3, 3, 3, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 0, 1, 3, 4, 3, 1, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 1, 0, 0, 7, 5, 7, 0, 0, 1, 0, 0, 1, 1, 0, 7, 6, 6, 7, 0, 1, 1, 0, 0, 0, 2, 0, 7, 7, 6, 7, 7, 0, 2, 0, 0

Offset: 0

Rémy Sigrist, Feb 21 2021

For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.

The underlying idea is to merge in an optimal way the runs in binary expansions of n and of k so that they match, hence the relationship with A003188.

			Array T(n, k) begins:
  n\k|  0  1  2  3  4  5  6  7   8   9  10  11  12  13  14  15
  ---+--------------------------------------------------------
    0|  0  0  0  0  0  0  0  0   0   0   0   0   0   0   0   0
    1|  0  1  1  0  0  1  1  0   0   1   1   0   0   1   1   0
    2|  0  1  2  3  3  2  1  0   0   1   2   3   3   2   1   0
    3|  0  0  3  3  3  3  0  0   0   0   3   3   3   3   0   0
    4|  0  0  3  3  4  4  7  7   7   7   4   4   3   3   0   0
    5|  0  1  2  3  4  5  6  7   7   6   5   4   3   2   1   0
    6|  0  1  1  0  7  6  6  7   7   6   6   7   0   1   1   0
    7|  0  0  0  0  7  7  7  7   7   7   7   7   0   0   0   0
    8|  0  0  0  0  7  7  7  7   8   8   8   8  15  15  15  15
    9|  0  1  1  0  7  6  6  7   8   9   9   8  15  14  14  15
   10|  0  1  2  3  4  5  6  7   8   9  10  11  12  13  14  15
   11|  0  0  3  3  4  4  7  7   8   8  11  11  12  12  15  15
   12|  0  0  3  3  3  3  0  0  15  15  12  12  12  12  15  15
   13|  0  1  2  3  3  2  1  0  15  14  13  12  12  13  14  15
   14|  0  1  1  0  0  1  1  0  15  14  14  15  15  14  14  15
   15|  0  0  0  0  0  0  0  0  15  15  15  15  15  15  15  15

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10
Rémy Sigrist, Colored representation of the table for n, k < 2^10 (black pixels correspond to 0's)
Rémy Sigrist, PARI program for A341840
Index entries for sequences related to binary expansion of n

Cf. A003188, A003987, A005811, A070939, A227736, A341839, A341840, A341841.

PARI
```
See Links section.
```

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, n) = n.
T(n, 0) = 0.
A070939(T(n, k)) <= min(A070939(n), A070939(k)).
A003188(T(n, k)) = A003188(n) AND A003188(k) (where AND denotes the bitwise AND operator).

A341839 Square array T(n, k), n, k >= 0, read by antidiagonals; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) is the union of R(n) and of R(k).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 4, 2, 2, 2, 4, 5, 5, 2, 2, 5, 5, 6, 5, 5, 3, 5, 5, 6, 7, 6, 5, 4, 4, 5, 6, 7, 8, 6, 5, 5, 4, 5, 5, 6, 8, 9, 9, 5, 5, 5, 5, 5, 5, 9, 9, 10, 9, 10, 4, 5, 5, 5, 4, 10, 9, 10, 11, 10, 10, 11, 4, 5, 5, 4, 11, 10, 10, 11, 12, 10, 10, 10, 11, 5, 6, 5, 11, 10, 10, 10, 12

Offset: 0

Rémy Sigrist, Feb 21 2021

For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.

The underlying idea is to break in an optimal way the runs in binary expansions of n and of k so that they match, hence the relationship with A003188.

			Array T(n, k) begins:
  n\k|    0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+-----------------------------------------------------------------
    0|    0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|    1   1   2   2   5   5   6   6   9   9  10  10  13  13  14  14
    2|    2   2   2   2   5   5   5   5  10  10  10  10  13  13  13  13
    3|    3   2   2   3   4   5   5   4  11  10  10  11  12  13  13  12
    4|    4   5   5   4   4   5   5   4  11  10  10  11  11  10  10  11
    5|    5   5   5   5   5   5   5   5  10  10  10  10  10  10  10  10
    6|    6   6   5   5   5   5   6   6   9   9  10  10  10  10   9   9
    7|    7   6   5   4   4   5   6   7   8   9  10  11  11  10   9   8
    8|    8   9  10  11  11  10   9   8   8   9  10  11  11  10   9   8
    9|    9   9  10  10  10  10   9   9   9   9  10  10  10  10   9   9
   10|   10  10  10  10  10  10  10  10  10  10  10  10  10  10  10  10
   11|   11  10  10  11  11  10  10  11  11  10  10  11  11  10  10  11
   12|   12  13  13  12  11  10  10  11  11  10  10  11  12  13  13  12
   13|   13  13  13  13  10  10  10  10  10  10  10  10  13  13  13  13
   14|   14  14  13  13  10  10   9   9   9   9  10  10  13  13  14  14
   15|   15  14  13  12  11  10   9   8   8   9  10  11  12  13  14  15

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10
Index entries for sequences related to binary expansion of n

Cf. A003188, A003987, A005811, A042963, A070939, A101211, A227736, A341840, A341841.

T(n,k) = { my (r=[], v=0); while (n||k, my (w=min(valuation(n+n%2,2), valuation(k+k%2,2))); r=concat(w,r); n\=2^w; k\=2^w); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }

T(n, k) = T(k, n)
T(m, T(n, k)) = T(T(m, n), k).
T(n, n) = n.
T(n, 0) = 0.
A070939(T(n, k)) = max(A070939(n), A070939(k)).
A003188(T(n, k)) = A003188(n) OR A003188(k) (where OR denotes the bitwise OR operator).
T(n, 1) = A042963(ceiling((n+1)/2)).

A341841 Square array T(n, k), n, k >= 0, read by antidiagonals upwards; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) equals R(n) minus R(k).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Feb 21 2021

For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.
This sequence uses set subtraction, and is related to:
- A003987 which uses set difference,
- A341839 which uses set union,
- A341840 which uses set intersection.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7  8  9  10  11  12  13  14  15
  ---+--------------------------------------------------------------
    0|   0   0   0   0   0   0   0   0  0  0   0   0   0   0   0   0
    1|   1   0   0   1   1   0   0   1  1  0   0   1   1   0   0   1
    2|   2   3   0   1   1   0   3   2  2  3   0   1   1   0   3   2
    3|   3   3   0   0   0   0   3   3  3  3   0   0   0   0   3   3
    4|   4   4   7   7   0   0   3   3  3  3   0   0   7   7   4   4
    5|   5   4   7   6   1   0   3   2  2  3   0   1   6   7   4   5
    6|   6   7   7   6   1   0   0   1  1  0   0   1   6   7   7   6
    7|   7   7   7   7   0   0   0   0  0  0   0   0   7   7   7   7
    8|   8   8   8   8  15  15  15  15  0  0   0   0   7   7   7   7
    9|   9   8   8   9  14  15  15  14  1  0   0   1   6   7   7   6
   10|  10  11   8   9  14  15  12  13  2  3   0   1   6   7   4   5
   11|  11  11   8   8  15  15  12  12  3  3   0   0   7   7   4   4
   12|  12  12  15  15  15  15  12  12  3  3   0   0   0   0   3   3
   13|  13  12  15  14  14  15  12  13  2  3   0   1   1   0   3   2
   14|  14  15  15  14  14  15  15  14  1  0   0   1   1   0   0   1
   15|  15  15  15  15  15  15  15  15  0  0   0   0   0   0   0   0

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10
Rémy Sigrist, PARI program for A341841
Index entries for sequences related to binary expansion of n

Cf. A003188, A003987, A070939, A227736, A341839, A341840.

PARI
```
See Links section.
```

T(n, n) = 0.
T(n, 0) = n.
T(T(n, k), k) = T(n, k).
A070939(T(n, k)) <= A070939(n).
A003188(T(n, k)) = A003188(n) - (A003188(n) AND A003188(k)) (where AND denotes the bitwise AND operator).

A335858 Nonnegative integers ordered by binary length and then lexicographically by run lengths (considering least significant runs first).

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 10, 13, 9, 14, 11, 12, 8, 15, 21, 26, 18, 29, 22, 25, 17, 30, 20, 27, 19, 28, 23, 24, 16, 31, 42, 53, 37, 58, 45, 50, 34, 61, 41, 54, 38, 57, 46, 49, 33, 62, 43, 52, 36, 59, 44, 51, 35, 60, 40, 55, 39, 56, 47, 48, 32, 63, 85, 106, 74

Offset: 0

Rémy Sigrist, Jun 27 2020

The variant where we consider most significant runs first apparently corresponds to A180200.

			The first terms, alongside the corresponding binary representation and run lengths, are:
  n   a(n)  bin(a(n))  A227736(n, *)
  --  ----  ---------  -------------
   0     0          0  ()
   1     1          1  (1)
   2     2         10  (1, 1)
   3     3         11  (2)
   4     5        101  (1, 1, 1)
   5     6        110  (1, 2)
   6     4        100  (2, 1)
   7     7        111  (3)
   8    10       1010  (1, 1, 1, 1)
   9    13       1101  (1, 1, 2)
  10     9       1001  (1, 2, 1)
  11    14       1110  (1, 3)
  12    11       1011  (2, 1, 1)
  13    12       1100  (2, 2)
  14     8       1000  (3, 1)
  15    15       1111  (4)

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program for A335858
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A005811, A180200, A227736.

PARI
```
See Links section.
```

Apparently a(n) = A056539(A180200(n)).

A339674 Irregular triangle T(n, k), n, k >= 0, read by rows; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; row n corresponds to the numbers k such that R(k) is included in R(n), in ascending order.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Feb 21 2021

For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.
The underlying idea is to take some or all of the rightmost runs of a number, and possibly merge some of them.
For any n >= 0, the n-th row:
- has 2^A000120(A003188(n)) terms,
- has first term 0 and last term A003817(n),
- has n at position A090079(n),
- corresponds to the distinct terms in n-th row of table A341840.

			The triangle starts:
    0;
    0, 1;
    0, 1, 2, 3;
    0, 3;
    0, 3, 4, 7;
    0, 1, 2, 3, 4, 5, 6, 7;
    0, 1, 6, 7;
    0, 7;
    0, 7, 8, 15;
    0, 1, 6, 7, 8, 9, 14, 15;
    0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15;
    0, 3, 4, 7, 8, 11, 12, 15;
    0, 3, 12, 15;
    0, 1, 2, 3, 12, 13, 14, 15;
    0, 1, 14, 15;
    0, 15;
    ...

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, Scatterplot of (n, T(n, k)) for n <= 2^10
Rémy Sigrist, PARI program for A339674
Index entries for sequences related to binary expansion of n

Cf. A000120, A003188, A003817, A090079, A227736, A341840.

PARI
```
See Links section.
```

T(n, 0) = 0.
T(n, A090079(n)) = n.
T(n, 2^A000120(A003188(n))-1) = A003817(n).

cp's OEIS Frontend

A227740 Integers from 0 to A037834(n) followed by integers from 0 to A037834(n+1) and so on.

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A163510 Irregular table read by rows: Write n in binary. For each 1, the m-th term of row n is the number of 0's between the m-th 1, reading right to left, and the (m-1)th 1 (or the right side of the number if m-1 = 0).

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A227741 Simple self-inverse permutation of natural numbers: List each block of A005811(n) numbers from A173318(n-1)+1 to A173318(n) in reverse order.

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A318927 Take the binary expansion of n, starting with the most significant bit, and concatenate the lengths of the runs.

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A318926 Take the binary expansion of n, starting with the least significant bit, and concatenate the lengths of the runs.

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A341840 Square array T(n, k), n, k >= 0, read by antidiagonals; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) is the intersection of R(n) and of R(k).

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A341839 Square array T(n, k), n, k >= 0, read by antidiagonals; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) is the union of R(n) and of R(k).

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