A334946 Irregular triangle read by rows: T(n,k) is the number of partitions of n into k consecutive parts that differ by 6, 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, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0

Offset

1

Comments

T(n,k) is 0 or 1, so T(n,k) represents the "existence" of the mentioned partition: 1 = exists, 0 = does not exist.

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

This is also an irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists 1'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 A303300.

Links

Table of n, a(n) for n=1..105.

Example

Triangle begins (rows 1..24).
1;
1;
1;
1;
1;
1;
1;
1, 1;
1, 0;
1, 1;
1, 0;
1, 1;
1, 0;
1, 1;
1, 0;
1, 1;
1, 0;
1, 1;
1, 0;
1, 1;
1, 0, 1;
1, 1, 0;
1, 0, 0;
1, 1, 1;
...
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], so the 24th row of this triangle is [1, 1, 1].

Crossrefs

Row sums give A334948.

Triangles of the same family where the parts differ by m are: A051731 (m=0), A237048 (m=1), A303300 (m=2), A330887 (m=3), A334460 (m=4), A334465 (m=5), this sequence (m=6).

Cf. A000567, A334947, A334949, A334953.

Sequence in context: A086299 A131364 A152066 * A225595 A228813 A122255

Adjacent sequences: A334943 A334944 A334945 * A334947 A334948 A334949

Keyword

nonn,tabf

Author

Omar E. Pol, May 27 2020

Status

approved