A334465 Irregular triangle read by rows: T(n,k) is the number of partitions of n into k consecutive parts that differ by 5, 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, 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, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1

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 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 A303300.

Links

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

Example

Triangle begins (rows 1..27).
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, 0, 0;
1, 1, 1;
1, 0, 0;
1, 1, 0;
1, 0, 1;
1, 1, 0;
1, 0, 0;
1, 1, 1;
...
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, 1, 1].

Maple

A334465 := proc(n, k)
    local first1 ;
    first1 := A000566(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>= A000566(k) then
            printf("%d, ", A334465(n, k)) ;
        else
            break;
        end if;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Oct 02 2020

Crossrefs

Row sums give A334541.

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). this sequence (m=5).

Cf. A000566.

Sequence in context: A273592 A279693 A368992 * A054527 A137794 A357731

Adjacent sequences: A334462 A334463 A334464 * A334466 A334467 A334468

Keyword

nonn,tabf

Author

Omar E. Pol, May 05 2020

Status

approved