A281011 Triangle read by rows in which row n lists the boundaries of the subparts of the symmetric representation of sigma(n). (See Comments for precise definition).

1, -1, 1, 0, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1

Offset

1

Comments

For the construction of this triangle we start with the diagram of A237048.

And then with the diagram of the isosceles triangle of A279693 as shown below:

Row _ _

1 _|1|1|_

2 _|1 _|_ 1|_

3 _|1 |1|1| 1|_

4 _|1 _|0|0|_ 1|_

5 _|1 |1 _|_ 1| 1|_

6 _|1 _|0|1|1|0|_ 1|_

7 _|1 |1 |0|0| 1| 1|_

8 _|1 _|0 _|0|0|_ 0|_ 1|_

9 _|1 |1 |1 _|_ 1| 1| 1|_

10 _|1 _|0 |0|1|1|0| 0|_ 1|_

11 _|1 |1 _|0|0|0|0|_ 1| 1|_

12 _|1 _|0 |1 |0|0| 1| 0|_ 1|_

13 _|1 |1 |0 _|0|0|_ 0| 1| 1|_

14 _|1 _|0 _|0|1 _|_ 1|0|_ 0|_ 1|_

15 _|1 |1 |1 |0|1|1|0| 1| 1| 1|_

16 |1 |0 |0 |0|0|0|0| 0| 0| 1|

...

Then filling the empty cells of the structure with zeros.

Then we replace the 1's that are associated to the even-indexed zig-zag paths in the left hand part of the structure with -1's.

Finally we replace the 1's that are associated to the odd-indexed zig-zag paths in the right hand part of the structure with -1's, as shown below:

Illustration of initial terms as an isosceles triangle:

Row

1 1,-1;

2 1, 0, 0,-1;

3 1, 0,-1, 1, 0,-1;

4 1, 0, 0, 0, 0, 0, 0,-1;

5 1, 0, 0,-1, 0, 0, 1, 0, 0,-1;

6 1, 0, 0, 0, 0, 1,-1, 0, 0, 0, 0,-1;

7 1, 0, 0, 0,-1, 0, 0, 0, 0, 1, 0, 0, 0,-1;

8 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1;

9 1, 0, 0, 0, 0,-1, 0, 1, 0, 0,-1, 0, 1, 0, 0, 0, 0,-1;

10 1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 0, 0, 0, 0, 0, 0,-1;

11 1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,-1;

12 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0,-1;

...

Note that the mentioned triangles are related to isosceles triangle A237593 and to the front view of the pyramid described in A245092.

The position of both the 1's and -1's in the n-th row of the diagram is related to the subparts of the symmetric representation of sigma(n).

The partial sums in row n, except the last term, gives the n-th row of A249351.

For more information about the "subparts" see A279387 and A281010.

Links

Table of n, a(n) for n=1..132.

Example

Triangle begins:
1,-1;
1, 0, 0,-1;
1, 0,-1, 1, 0,-1;
1, 0, 0, 0, 0, 0, 0,-1;
1, 0, 0,-1, 0, 0, 1, 0, 0,-1;
1, 0, 0, 0, 0, 1,-1, 0, 0, 0, 0,-1;
1, 0, 0, 0,-1, 0, 0, 0, 0, 1, 0, 0, 0,-1;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1;
1, 0, 0, 0, 0,-1, 0, 1, 0, 0,-1, 0, 1, 0, 0, 0, 0,-1;
1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 0, 0, 0, 0, 0, 0,-1;
1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,-1;
1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0,-1;
...

Crossrefs

Absolute values give A275601.

Row n is also the row 2n of A281010.

Row sums gives A000004.

Cf. A196020, A235791, A236194, A237048, A237270, A237591, A237593, A244050, A245092, A249351, A262626.

Sequence in context: A091219 A304109 A275601 * A016416 A014024 A014039

Adjacent sequences: A281008 A281009 A281010 * A281012 A281013 A281014

Keyword

sign,tabf

Author

Omar E. Pol, Jan 14 2017

Status

approved