A281010 Triangle read by rows in which row 2n-1 lists the widths of the symmetric representation of sigma(n), and row 2n lists a finite sequence S together with -1, with the property that the partial sums of S give the row 2n-1.

1, 1, -1, 1, 1, 1, 1, 0, 0, -1, 1, 1, 0, 1, 1, 1, 0, -1, 1, 0, -1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset

1,61

Comments

The row 2n-1 lists the widths of the terraces at the n-th level (starting from the top) of the pyramid described in A245092.

The sum of the areas of these terraces equals A000203(n): the sum of the divisors of n.

The k-th element of row 2n is associated to the k-th vertical cells at the n-th level of the pyramid.

The row 2n shows where the subparts (or subregions) of the terraces starting and ending, in accordance with the values 1 or -1.

The number of subparts in the n-th terrace equals A001227(n): the number of odd divisors of n.

If n is odd then the number of subparts in the n-th terrace is also A000005(n): the number of divisors of n.

Links

Table of n, a(n) for n=1..136.

Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)

Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Example

Triangle begins:
1;
1,-1;
1, 1, 1;
1, 0, 0,-1;
1, 1, 0, 1, 1;
1, 0,-1, 1, 0;-1;
1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 0,-1;
1, 1, 1, 0, 0, 0, 1, 1, 1;
1, 0, 0,-1, 0, 0, 1, 0, 0,-1;
1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 1,-1, 0, 0, 0, 0,-1;
1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1;
1, 0, 0, 0,-1, 0, 0, 0, 0, 1, 0, 0, 0,-1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-1;
1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0,-1, 0, 1, 0, 0,-1, 0, 1, 0, 0, 0, 0,-1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 0, 0, 0, 0, 0, 0,-1;
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1;
1, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,-1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 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;
...
Written as an isosceles triangle the sequence begins:
.
.                                        1;
.                                      1, -1;
.                                    1,  1,  1;
.                                  1,  0,  0, -1;
.                                1,  1,  0,  1,  1;
.                              1,  0, -1,  1,  0, -1;
.                            1,  1,  1,  1,  1,  1,  1;
.                          1,  0,  0,  0,  0,  0,  0, -1;
.                        1,  1,  1,  0,  0,  0,  1,  1,  1;
.                      1,  0,  0, -1,  0,  0,  1,  0,  0, -1;
.                    1,  1,  1,  1,  1,  2,  1,  1,  1,  1,  1;
.                  1,  0,  0,  0,  0,  1, -1,  0,  0,  0,  0, -1;
.                1,  1,  1,  1,  0,  0,  0,  0,  0,  1,  1,  1,  1;
.              1,  0,  0,  0, -1,  0,  0,  0,  0,  1,  0,  0,  0, -1;
.            1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1;
.          1,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, -1;
.        1,  1,  1,  1,  1,  0,  0,  1,  1,  1,  0,  0,  1,  1,  1,  1,  1;
.      1,  0,  0,  0,  0, -1,  0,  1,  0,  0, -1,  0,  1,  0,  0,  0,  0, -1;
.    1,  1,  1,  1,  1,  1,  1,  1,  1,  0,  1,  1,  1,  1,  1,  1,  1,  1,  1;
.  1,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1,  0,  0,  0,  0,  0,  0,  0,  0, -1;
...

Crossrefs

The sum of row 2n-1 is A000203(n).

The sum of row 2n is A000004(n) = 0.

The number of positive terms in row 2n is A001227(n).

The number of nonzero terms in row 2n is A054844(n).

Middle diagonal (or central column of the isosceles triangle) gives A067742.

Row 2n-1 is also the n-th row of A249351.

Row 2n is also the n-th row of A281011.

Row 2n-1 lists the partial sums of the terms, except the last term, of the row 2n.

Cf. A000005, A001227, A196020, A235048, A236104, A237048, A237591, A237593, A244050, A245092, A262626, A279387, A279388, A279391.

Sequence in context: A330261 A157746 A349082 * A316864 A037820 A076493

Adjacent sequences: A281007 A281008 A281009 * A281011 A281012 A281013

Keyword

sign,tabl

Author

Omar E. Pol, Jan 12 2017

Status

approved