login
A347579
Irregular triangle read by rows: T(n,k) = 1 iff there is no partition of n into exactly k consecutive parts, for n >= 1, 1 <= k <= A003056(n).
2
0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1
OFFSET
1
COMMENTS
The connection with A196020 is as follow: this sequence --> A237048 --> A235791 --> A236104 --> A196020.
FORMULA
T(n,k) = 1 - A237048(n,k).
EXAMPLE
Triangle begins (rows 1..28):
0;
0;
0, 0;
0, 1;
0, 0;
0, 1, 0;
0, 0, 1;
0, 1, 1;
0, 0, 0;
0, 1, 1, 0;
0, 0, 1, 1;
0, 1, 0, 1;
0, 0, 1, 1;
0, 1, 1, 0;
0, 0, 0, 1, 0;
0, 1, 1, 1, 1;
0, 0, 1, 1, 1;
0, 1, 0, 0, 1;
0, 0, 1, 1, 1;
0, 1, 1, 1, 0;
0, 0, 0, 1, 1, 0;
0, 1, 1, 0, 1, 1;
0, 0, 1, 1, 1, 1;
0, 1, 0, 1, 1, 1;
0, 0, 1, 1, 0, 1;
0, 1, 1, 0, 1, 1;
0, 0, 0, 1, 1, 0;
0, 1, 1, 1, 1, 1, 0;
...
For n = 15 the partitions of 15 into consecutive parts are [15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1]. We can see that a partition of 15 into four consecutive parts does not exist, hence there is a "nonexistence" of such partition, so T(15,4) = 1.
Illustration of initial terms:
Row _
1 _|0|
2 _|0 _|
3 _|0 |0|
4 _|0 _|1|
5 _|0 |0 _|
6 _|0 _|1|0|
7 _|0 |0 |1|
8 _|0 _|1 _|1|
9 _|0 |0 |0 _|
10 _|0 _|1 |1|0|
11 _|0 |0 _|1|1|
12 _|0 _|1 |0 |1|
13 _|0 |0 |1 _|1|
14 _|0 _|1 _|1|0 _|
15 _|0 |0 |0 |1|0|
16 _|0 _|1 |1 |1|1|
17 _|0 |0 _|1 _|1|1|
18 _|0 _|1 |0 |0 |1|
19 _|0 |0 |1 |1 _|1|
20 _|0 _|1 _|1 |1|0 _|
21 _|0 |0 |0 _|1|1|0|
22 _|0 _|1 |1 |0 |1|1|
23 _|0 |0 _|1 |1 |1|1|
24 _|0 _|1 |0 |1 _|1|1|
25 _|0 |0 |1 _|1|0 |1|
26 _|0 _|1 _|1 |0 |1 _|1|
27 _|0 |0 |0 |1 |1|0 _|
28 |0 |1 |1 |1 |1|1|0|
...
For the connection between the above diagram and the partitions of n into consecutive parts see the diagrams of A286000 and A286001.
CROSSREFS
Row sums give A238005.
Row n has length A003056(n).
Column k starts in row A000217(k).
The number of zeros in row n equals A001227(n).
Sequence in context: A358775 A354981 A231600 * A280710 A354353 A353669
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Sep 07 2021
STATUS
approved