login
A339275
Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists the terms of A040000: 1, 2, 2, 2, ... interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.
3
1, 2, 2, 1, 2, 0, 2, 2, 2, 0, 1, 2, 2, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 1, 2, 2, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 1, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2, 2, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 2
OFFSET
1,2
COMMENTS
T(n,k) is also the number of horizontal line segments in the n-th level of the k-th largest double-staircase of the diagram defined in A335616 (see example).
The partial sums of column k give the k-th column of A338721.
EXAMPLE
Triangle begins (rows 1..28):
1;
2;
2, 1;
2, 0;
2, 2;
2, 0, 1;
2, 2, 0;
2, 0, 0;
2, 2, 2;
2, 0, 0, 1;
2, 2, 0, 0;
2, 0, 2, 0;
2, 2, 0, 0;
2, 0, 0, 2;
2, 2, 2, 0, 1;
2, 0, 0, 0, 0;
2, 2, 0, 0, 0;
2, 0, 2, 2, 0;
2, 2, 0, 0, 0;
2, 0, 0, 0, 2;
2, 2, 2, 0, 0, 1;
2, 0, 0, 2, 0, 0;
2, 2, 0, 0, 0, 0;
2, 0, 2, 0, 0, 0;
2, 2, 0, 0, 2, 0;
2, 0, 0, 2, 0, 0;
2, 2, 2, 0, 0, 2;
2, 0, 0, 0, 0, 0, 1;
...
For an illustration of the rows of triangle consider the infinite "double-staircases" diagram defined in A335616.
The first 15 levels of the structure looks like this:
.
Level "Double-staircases" diagram
n _
1 _|1|_
2 _|1 _ 1|_
3 _|1 |1| 1|_
4 _|1 _| |_ 1|_
5 _|1 |1 _ 1| 1|_
6 _|1 _| |1| |_ 1|_
7 _|1 |1 | | 1| 1|_
8 _|1 _| _| |_ |_ 1|_
9 _|1 |1 |1 _ 1| 1| 1|_
10 _|1 _| | |1| | |_ 1|_
11 _|1 |1 _| | | |_ 1| 1|_
12 _|1 _| |1 | | 1| |_ 1|_
13 _|1 |1 | _| |_ | 1| 1|_
14 _|1 _| _| |1 _ 1| |_ |_ 1|_
15 |1 |1 |1 | |1| | 1| 1| 1|
.
For n = 15, in the 15th level of the diagram we have that the first largest double-staircase has two horizontal steps, the second double-staircase has two steps, the third double-staircase has two steps, there are no steps in the fourth double-stairce and the fifth double-staircase has only one step, so the 15th row of triangle is [2, 2, 2, 0, 1].
CROSSREFS
Column 1 is A040000.
Row sums give A335616.
Row n has length A003056(n).
Column k starts in row A000217(k).
The number of positive terms in row n is A001227(n).
Sequence in context: A156381 A089077 A203398 * A225064 A361967 A130071
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 01 2020
STATUS
approved