A341839 -id:A341839 - cp's OEIS frontend

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).

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).

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).

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula