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).

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

Author

Rémy Sigrist, Feb 21 2021

Keywords

• nonn
• tabl
• base

Comments

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.

Examples

			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
		

Links

• 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

Crossrefs

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

Programs

PARI

See Links section.

Formula

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).