A334540 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 5, n >= 1, k >= 1, and the first element of column k is in the row that is the k-th heptagonal number (A000566).

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

Offset

1,8

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 heptagonal 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..105.

Formula

T(n,k) = k*A334465(n,k).

Example

Triangle begins (rows 1..27):
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, 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;
...
For n = 27 there are three partitions of 27 into consecutive parts that differ by 5, including 27 as a valid partition. They are [27], [16, 11] and [14, 9, 4]. The number of parts of these partitions are 1, 2, 3 respectively, so the 27th row of the triangle is [1, 2, 3].

Maple

A334540 := proc(n, k)
        k*A334465(n, k) ;
end proc: # R. J. Mathar, Oct 02 2020

Crossrefs

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), this sequence (d=5).

Cf. A000566, A334465.

Sequence in context: A230074 A230075 A334947 * A334462 A054848 A284171

Adjacent sequences: A334537 A334538 A334539 * A334541 A334542 A334543

Keyword

nonn,tabf

Author

Omar E. Pol, May 05 2020

Status

approved