A334460 Irregular triangle read by rows: T(n,k) is the number of partitions of n into k consecutive parts that differ by 4, and the first element of column k is in the row that is the k-th hexagonal number (A000384).

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, 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, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 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 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 hexagonal 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..106.

Example

Triangle begins (rows 1..28):
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, 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;
...
For n = 28 there are three partitions of 28 into consecutive parts that differ by 4, including 28 as a partition. They are [28], [16, 12] and [13, 9, 5, 1]. The number of parts of these partitions are 1, 2, 4 respectively. There are no partitions of this kind with three parts, so the 28th row of the triangle is [1, 1, 0, 1].

Maple

A334460 := proc(n, k)
    local first1 ;
    first1 := A000384(k) ;
    if n < first1 then
        0 ;
    elif modp(n-first1, k) = 0 then
        1;
    else
        0;
    end if;
end proc:
for n from 1 to 40 do
    for k from 1 do
        if n>= A000384(k) then
            printf("%d, ", A334460(n, k)) ;
        else
            break;
        end if;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Oct 02 2020

Crossrefs

Row sums give A334461.

Triangles of the same family where the parts differ by d are A051731 (d=0), A237048 (d=1), A303300 (d=2), A330887 (d=3), this sequence (d=4).

Cf. A000384, A327262, A334462.

Sequence in context: A351824 A365605 A365716 * A071023 A166280 A340371

Adjacent sequences: A334457 A334458 A334459 * A334461 A334462 A334463

Keyword

nonn,tabf

Author

Omar E. Pol, May 01 2020

Status

approved