A262611 Triangle read by rows in which row n lists the widths of the symmetric representation of A024916(n): the sum of all divisors of all positive integers <= n.

1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 3, 2, 1, 1, 2, 3, 3, 3, 3, 3, 2, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 2, 3, 4, 4, 4, 5, 4, 4, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 5, 6, 5, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 5, 6, 7, 6, 5, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 6, 7, 7, 7, 6, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 6, 6, 7, 7, 7, 6, 6, 6, 6, 5, 4, 3, 2, 1

Offset

1,3

Comments

Here T(n,k) is defined to be the "k-th width" of the symmetric representation of A024916(n), with n>=1 and 1<=k<=2n-1.

If both A249351 and this sequence are written as isosceles triangles then the partial sums of the columns of A249351 give the columns of this isosceles triangle (see the second triangle in Example section).

For the definition of the k-th width of the symmetric representation of sigma(n) see A249351.

Note that for the geometric representation of the n-th row of the triangle we need the x-axis, the y-axis, and only a Dyck path which is given by the elements of the n-th row of the triangle A237593.

Row n has length 2*n-1.

Row sums give A024916.

The middle diagonal is A240542.

Links

Table of n, a(n) for n=1..121.

Example

Triangle begins:
1;
1,2,1;
1,2,2,2,1;
1,2,3,3,3,2,1;
1,2,3,3,3,3,3,2,1;
1,2,3,4,4,5,4,4,3,2,1;
1,2,3,4,4,4,5,4,4,4,3,2,1;
1,2,3,4,5,5,5,6,5,5,5,4,3,2,1;
1,2,3,4,5,5,5,6,7,6,5,5,5,4,3,2,1;
1,2,3,4,5,6,6,6,7,7,7,6,6,6,5,4,3,2,1;
1,2,3,4,5,6,6,6,6,7,7,7,6,6,6,6,5,4,3,2,1;
1,2,3,4,5,6,7,7,7,8,9,9,9,8,7,7,7,6,5,4,3,2,1;
...
--------------------------------------------------------------------------
.        Written as an isosceles triangle
.              the sequence begins:               Diagram for n = 1..12
--------------------------------------------------------------------------
.                                                _ _ _ _ _ _ _ _ _ _ _ _
.                      1;                       |_| | | | | | | | | | | |
.                    1,2,1;                     |_ _|_| | | | | | | | | |
.                  1,2,2,2,1;                   |_ _|  _|_| | | | | | | |
.                1,2,3,3,3,2,1;                 |_ _ _|    _|_| | | | | |
.              1,2,3,3,3,3,3,2,1;               |_ _ _|  _|  _ _|_| | | |
.            1,2,3,4,4,5,4,4,3,2,1;             |_ _ _ _|  _| |  _ _|_| |
.          1,2,3,4,4,4,5,4,4,4,3,2,1;           |_ _ _ _| |_ _|_|    _ _|
.        1,2,3,4,5,5,5,6,5,5,5,4,3,2,1;         |_ _ _ _ _|  _|     |
.      1,2,3,4,5,5,5,6,7,6,5,5,5,4,3,2,1;       |_ _ _ _ _| |      _|
.    1,2,3,4,5,6,6,6,7,7,7,6,6,6,5,4,3,2,1;     |_ _ _ _ _ _|  _ _|
.  1,2,3,4,5,6,6,6,6,7,7,7,6,6,6,6,5,4,3,2,1;   |_ _ _ _ _ _| |
.1,2,3,4,5,6,7,7,7,8,9,9,9,8,7,7,7,6,5,4,3,2,1; |_ _ _ _ _ _ _|
...
For n = 3 the symmetric representation of A024916(3) = 8 in the 4th quadrant looks like this:
.
.    Polygon         Cells
.     _ _ _          _ _ _
.    |     |        |_|_|_|
.    |    _|        |_|_|_|
.    |_ _|          |_|_|
.
There are eight cells. The representation of the widths looks like this:
.
.     \ \ \
.     \ \ \
.     \ \    1
.          2 2
.        1 2
.
So the third row of the triangle is [1, 2, 2, 2, 1].

Crossrefs

Cf. A024916, A236104, A237048, A237593, A240542, A241008, A241010, A244250, A245685, A246955, A247687, A249351, A250068, A250070, A250071, A253258, A262626.

Sequence in context: A053874 A170906 A123245 * A110535 A033941 A053256

Adjacent sequences: A262608 A262609 A262610 * A262612 A262613 A262614

Keyword

nonn,tabf

Author

Omar E. Pol, Sep 26 2015

Status

approved