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).

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.

Links

Table of n, a(n) for n=1..94.

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).

Cf. A196020, A235791, A236104, A237048, A237593, A280850, A286000, A286001, A296508.

Sequence in context: A358775 A354981 A231600 * A280710 A354353 A353669

Adjacent sequences: A347576 A347577 A347578 * A347580 A347581 A347582

Keyword

nonn,tabf

Author

Omar E. Pol, Sep 07 2021

Status

approved