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

0,4

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.

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

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).

Example

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

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A195664 A053202 A188122 * A050186 A334218 A342984

Adjacent sequences: A341838 A341839 A341840 * A341842 A341843 A341844

Keyword

nonn,tabl,base

Author

Rémy Sigrist, Feb 21 2021

Status

approved