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.

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

Author

Rémy Sigrist, Feb 21 2021

Keywords

• nonn
• base
• tabf

Comments

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.

Examples

			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;
    ...
		

Links

• 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

Crossrefs

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

Programs

PARI

See Links section.

Formula

T(n, 0) = 0.

T(n, A090079(n)) = n.

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