A334947 Irregular triangle read by rows: T(n,k) is the number of parts in the partition of n into k consecutive parts that differ by 6, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th octagonal number (A000567).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 0, 1, 2, 3, 1, 0, 0, 1, 2, 0, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 1, 2, 3, 0, 1, 0, 0, 0

Offset

1,9

Comments

Since the trivial partition n is counted, so T(n,1) = 1.

This is an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k's interleaved with k-1 zeros, and the first element of column k is in the row that is the k-th octagonal number.

This triangle can be represented with a diagram of overlapping curves, in which every column of triangle is represented by a periodic curve.

For a general theorem about the triangles of this family see A285914.

Links

Table of n, a(n) for n=1..106.

Formula

T(n,k) = k*A334946(n,k).

Example

Triangle begins (rows 1..24).
1;
1;
1;
1;
1;
1;
1;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0;
1, 2;
1, 0, 3;
1, 2, 0;
1, 0, 0;
1, 2, 3;
...
For n = 24 there are three partitions of 24 into consecutive parts that differ by 6, including 24 as a valid partition. They are [24], [15, 9] and [14, 8, 2]. There are 1, 2 and 3 parts respectively, so the 24th row of this triangle is [1, 2, 3].

Crossrefs

Row sums give A334949.

Triangles of the same family where the parts differ by d are A127093 (d=0), A285914 (d=1), A330466 (d=2), A330888 (d=3), A334462 (d=4), A334540 (d=5), this sequence (d=6).

Cf. A000567, A334946, A334948, A334953.

Sequence in context: A118822 A230074 A230075 * A334540 A334462 A054848

Adjacent sequences: A334944 A334945 A334946 * A334948 A334949 A334950

Keyword

nonn,tabf

Author

Omar E. Pol, May 27 2020

Status

approved