A227736 -id:A227736 - cp's OEIS frontend

A228528 Triangle read by rows in which row n lists the compositions (ordered partitions) of n (see Comments lines for definition).

1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 4, 4, 1, 1, 3, 1, 2, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 3, 2, 1, 1, 3, 2, 3, 1, 4, 5, 5, 1, 1, 4, 1

Offset: 1

Omar E. Pol, Sep 07 2013

In order to construct this sequence we use the following rules: T(1,1) = 1. For n >= 2, row n lists the last 2^(n-2) compositions from the n-th row of triangle A228351 and then the last 2^(n-2) compositions from the n-th row of triangle A228525. In both cases these compositions are listed in the same order as they are listed in the mentioned triangles.
Row n has length A001792(n-1).
Row sums give A001787, n >= 1.
First differs from A227736 at a(18).

			Illustration of initial terms:
---------------------------------------
n   j     Diagram        Composition j
---------------------------------------
.          _
1   1     |_|_           1;
2   1     |_| |          1, 1,
2   2     |_ _|_         2;
3   1     |_  | |        2, 1,
3   2     |_|_| |        1, 1, 1,
3   3     |_|   |        1, 2,
3   4     |_ _ _|_       3;
4   1     |_    | |      3, 1,
4   2     |_|_  | |      1, 2, 1,
4   3     |_  | | |      2, 1, 1,
4   4     |_|_|_| |      1, 1, 1, 1,
4   5     |_| |   |      1, 1, 2,
4   6     |_ _|   |      2, 2,
4   7     |_|     |      1, 3,
4   8     |_ _ _ _|_     4;
5   1     |_      | |    4, 1,
5   2     |_|_    | |    1, 3, 1,
5   3     |_  |   | |    2, 2, 1,
5   4     |_|_|_  | |    1, 1, 2, 1,
5   5     |_    | | |    3, 1, 1,
5   6     |_|_  | | |    1, 2, 1, 1,
5   7     |_  | | | |    2, 1, 1, 1,
5   8     |_|_|_|_| |    1, 1, 1, 1, 1,
5   9     |_| | |   |    1, 1, 1, 2,
5  10     |_ _| |   |    2, 1, 2,
5  11     |_|   |   |    1, 2, 2,
5  12     |_ _ _|   |    3, 2,
5  13     |_| |     |    1, 1, 3,
5  14     |_ _|     |    2, 3,
5  15     |_|       |    1, 4,
5  16     |_ _ _ _ _|    5;
.
Triangle begins:
[1];
[1,1], [2];
[2,1], [1,1,1], [1,2], [3];
[3,1], [1,2,1], [2,1,1], [1,1,1,1], [1,1,2], [2,2], [1,3], [4];
[4,1], [1,3,1], [2,2,1], [1,1,2,1], [3,1,1], [1,2,1,1], [2,1,1,1], [1,1,1,1,1], [1,1,1,2], [2,1,2], [1,2,2], [3,2], [1,1,3], [2,3], [1,4], [5];

Cf. A001792, A001787, A011782, A066099, A227736, A228525, A228351, A228369.

cp's OEIS Frontend

A228528 Triangle read by rows in which row n lists the compositions (ordered partitions) of n (see Comments lines for definition).

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