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.
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
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, May 01 2020
STATUS
approved