A261350 Triangle read by rows T(n,k) which is the mirror of A237591.

1, 2, 1, 2, 1, 3, 2, 3, 1, 1, 4, 1, 2, 4, 1, 2, 5, 2, 2, 5, 1, 1, 2, 6, 1, 1, 3, 6, 1, 2, 2, 7, 1, 2, 3, 7, 2, 1, 3, 8, 1, 1, 2, 3, 8, 1, 1, 2, 3, 9, 1, 1, 2, 4, 9, 1, 2, 2, 3, 10, 1, 2, 2, 4, 10, 2, 1, 2, 4, 11, 1, 1, 1, 3, 4, 11, 1, 1, 2, 2, 4, 12, 1, 1, 2, 2, 5, 12, 1, 1, 2, 3, 4, 13, 1, 2, 1, 3, 5, 13, 1, 2, 2, 2, 5, 14

Offset: 1

Author

Omar E. Pol, Aug 18 2015

Keywords

• nonn
• tabf

Comments

Row n has length A003056(n) hence column k starts in row A000217(k).

Row sums give A000027.

Right border gives A008619, n >= 1.

n is an odd prime if and only if T(n,r-1) = 1 + T(n-1,r-1) and T(n,k) = T(n-1,k) for the rest of the values of k, where r = A003056(n) is the number of elements in row n.

T(n,k) is also the length of the k-th segment in a zig-zag path on the first quadrant of the square grid, connecting the point (m, m) with the point (0, n), ending with a segment in horizontal direction, where m = A240542(n). The area of the polygon defined by the y-axis, this zig-zag path and the diagonal [(0, 0), (m, m)], is equal to A024916(n)/2, one half of the sum of all divisors of all positive integers <= n. Therefore the reflected polygon, which is adjacent to the x-axis, with the zig-zag path connecting the point (n, 0) with the point (m, m), has the same property. And so on for each octant in the four quadrants.

For the representation of A024916 and A000203 we use two octants, for example: the first octant and the second octant, or the 6th octant and the 7th octant, etc., see A237593.

The elements of the n-th row of A237591 together with the elements of the n-th row of this sequence give the n-th row of A237593.

The connection between A196020 and A237271 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> this sequence --> A237593 --> A239660 --> A237270 --> A237271.

T(n,k) is also the area (or the number of cells) of the k-th vertical side at the n-th level (starting from the top) in the right part of the front view of the stepped pyramid described in A245092, see Example section.

Examples

			Triangle begins:
Row
1                       1;
2                       2;
3                    1, 2;
4                    1, 3;
5                    2, 3;
6                 1, 1, 4;
7                 1, 2, 4;
8                 1, 2, 5;
9                 2, 2, 5;
10             1, 1, 2, 6;
11             1, 1, 3, 6;
12             1, 2, 2, 7;
13             1, 2, 3, 7;
14             2, 1, 3, 8;
15          1, 1, 2, 3, 8;
16          1, 1, 2, 3, 9;
17          1, 1, 2, 4, 9;
18          1, 2, 2, 3, 10;
19          1, 2, 2, 4, 10;
20          2, 1, 2, 4, 11;
21       1, 1, 1, 3, 4, 11;
22       1, 1, 2, 2, 4, 12;
23       1, 1, 2, 2, 5, 12;
24       1, 1, 2, 3, 4, 13;
25       1, 2, 1, 3, 5, 13;
26       1, 2, 2, 2, 5, 14;
...
Illustration of initial terms:
Row      _
1       |1|_
2       |_ 2|_
3       |1|  2|_
4       |1|_   3|_
5       |_ 2|    3|_
6       |1|1|_     4|_
7       |1|  2|      4|_
8       |1|_ 2|_       5|_
9       |_ 2|  2|        5|_
10      |1|1|  2|_         6|_
11      |1|1|_   3|          6|_
12      |1|  2|  2|_           7|_
13      |1|_ 2|    3|            7|_
14      |_ 2|1|_   3|_             8|_
15      |1|1|  2|    3|              8|_
16      |1|1|  2|    3|_               9|_
17      |1|1|_ 2|_     4|                9|_
18      |1|  2|  2|    3|_                10|_
19      |1|_ 2|  2|      4|                 10|_
20      |_ 2|1|  2|_     4|_                  11|_
21      |1|1|1|_   3|      4|                   11|_
22      |1|1|  2|  2|      4|_                    12|_
23      |1|1|  2|  2|_       5|                     12|_
24      |1|1|_ 2|    3|      4|_                      13|_
25      |1|  2|1|_   3|        5|                       13|_
26      |1|  2|  2|  2|        5|                         14|
...
Also the diagram represents the right part of the front view of the pyramid described in A245092. For the other half front view see A237591. For more information about the pyramid and the symmetric representation of sigma see A237593.
		

Crossrefs

Cf. A000203, A000217, A003056, A008016, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244580, A245092, A249351, A259176, A259177, A259179.