

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.


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,6


COMMENTS

For any m > 0, R(m) contains the partial sums of the mth 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 nth 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 nth 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



