A272120 Square array T(n,k), n>=1, k>=1, read by antidiagonals downwards in which column k lists the alternating row sums of the first k columns of the triangle A196020.

1, 1, 3, 1, 3, 5, 1, 3, 4, 7, 1, 3, 4, 7, 9, 1, 3, 4, 7, 6, 11, 1, 3, 4, 7, 6, 11, 13, 1, 3, 4, 7, 6, 12, 8, 15, 1, 3, 4, 7, 6, 12, 8, 15, 17, 1, 3, 4, 7, 6, 12, 8, 15, 10, 19, 1, 3, 4, 7, 6, 12, 8, 15, 13, 19, 21, 1, 3, 4, 7, 6, 12, 8, 15, 13, 19, 12, 23, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 23, 25, 1, 3, 4, 7

Offset: 1

Author

Omar E. Pol, Apr 20 2016

Keywords

• nonn
• tabl

Comments

Every column of this square array is associated to an isosceles triangle and to a stepped pyramid in the same way as the sequence A196020 is associated to the isosceles triangle of A237593 and to the pyramid described in A245092. Hence there are infinitely many isosceles triangles and infinitely many pyramids that are associated to this sequence.

In the Example section appears the triangles and the top views of the pyramids associated to the columns 1 and 2.

The sequence A196020 is associated to the isosceles triangle of A237593 as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593. Then the structure of the pyramid described in A245092 arises after the 90-degree-zig-zag folding of every row of the isosceles triangle of A237593.

Note that the first m terms of column k are also the first m terms of A000203, where m = A000217(k) + k = A000217(k+1) - 1 = A000096(k).

Examples

			The corner of the square array begins:
1,   1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1...
3,   3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3,  3...
5,   4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4,  4...
7,   7,  7,  7,  7,  7,  7,  7,  7,  7,  7,  7,  7,  7,  7...
9,   6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6...
11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12...
13,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8,  8...
15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15...
17, 10, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13...
19, 19, 19, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18...
21, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12...
23, 23, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28...
25, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14...
27, 27, 27, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24...
29, 16, 23, 23, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24...
...
For k = 1 the first two terms of column k are also the first two terms of A000203, i.e., [1, 3].
For k = 2 the first five terms of column k are also the first five terms of A000203, i.e., [1, 3, 4, 7, 6].
For k = 3 the first nine terms of column k are also the first nine terms of A000203, i.e., [1, 3, 4, 7, 6, 12, 8, 15, 13].
More generally, the first A000096(k) terms of column k are also the first A000096(k) terms of A000203.
.
Illustration of initial terms of the column 1:
.
.                   2D                                3D
.           Isosceles triangle             Top view of the pyramid
.             before folding                    after folding
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
n                  _|_                   T(n,1) _ _ _ _ _ _ _ _ _ x
1                _|_|_|_                    1  |_| | | | | | | |
2        y     _|_ _|_ _|_     x            3  |_ _| | | | | | |
3            _|_ _ _|_ _ _|_                5  |_ _ _| | | | | |
4          _|_ _ _ _|_ _ _ _|_              7  |_ _ _ _| | | | |
5        _|_ _ _ _ _|_ _ _ _ _|_            9  |_ _ _ _ _| | | |
6      _|_ _ _ _ _ _|_ _ _ _ _ _|_         11  |_ _ _ _ _ _| | |
7    _|_ _ _ _ _ _ _|_ _ _ _ _ _ _|_       13  |_ _ _ _ _ _ _| |
8   |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _|      15  |_ _ _ _ _ _ _ _|
.                                              |
.                                              y
.
Illustration of initial terms of the column 2:
.
.                   2D                                3D
.           Isosceles triangle             Top view of the pyramid
.             before folding                    after folding
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
n                  _|_                   T(n,2) _ _ _ _ _ _ _ _ _ x
1                _|_|_|_                    1  |_| | | | | | | |
2        y     _|_ _|_ _|_     x            3  |_ _|_| | | | | |
3            _|_ _|_|_|_ _|_                4  |_ _|  _|_| | | |
4          _|_ _ _|_|_|_ _ _|_              7  |_ _ _|  _ _|_| |
5        _|_ _ _|_ _|_ _|_ _ _|_            6  |_ _ _| |  _ _ _|
6      _|_ _ _ _|_ _|_ _|_ _ _ _|_         11  |_ _ _ _| |
7    _|_ _ _ _|_ _ _|_ _ _|_ _ _ _|_        8  |_ _ _ _| |
8   |_ _ _ _ _|_ _ _|_ _ _|_ _ _ _ _|      15  |_ _ _ _ _|
.                                              |
.                                              y
.
Illustration of initial terms of the column 3:
.
.                   2D                                3D
.           Isosceles triangle             Top view of the pyramid
.             before folding                    after folding
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
n                  _|_                   T(n,3) _ _ _ _ _ _ _ _ _ x
1                _|_|_|_                    1  |_| | | | | | | |
2        y     _|_ _|_ _|_     x            3  |_ _|_| | | | | |
3            _|_ _|_|_|_ _|_                4  |_ _|  _|_| | | |
4          _|_ _ _|_|_|_ _ _|_              7  |_ _ _|    _|_| |
5        _|_ _ _|_ _|_ _|_ _ _|_            6  |_ _ _|  _|  _ _|
6      _|_ _ _ _|_|_|_|_|_ _ _ _|_         12  |_ _ _ _|  _|
7    _|_ _ _ _|_ _|_|_|_ _|_ _ _ _|_        8  |_ _ _ _| |
8   |_ _ _ _ _|_ _|_|_|_ _|_ _ _ _ _|      15  |_ _ _ _ _|
.                                              |
.                                              y
.
		

Links

• Omar E. Pol, Illustration of first 28 rows of the isosceles triangle associated to the column k when k tends to infinity
• Omar E. Pol, Perspective view of the first 16 levels of the pyramid associated to the column k when k tends to infinity

Crossrefs

Column 1 is A005408.

Every diagonal starting with 1 gives A000203.

Columns converge to A000203.

Compare A245093.

Cf. A000096, A000217, A000290, A000330, A024916, A175254, A196020, A235791, A236104, A237591, A237593, A237270, A237271, A244050, A245092, A262626.