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

0,6

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.

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

Formula

T(n, 0) = 0.

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

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

Example

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;
    ...

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A059283 A160202 A195673 * A241070 A128621 A132385

Adjacent sequences: A339671 A339672 A339673 * A339675 A339676 A339677

Keyword

nonn,base,tabf

Author

Rémy Sigrist, Feb 21 2021

Status

approved