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.
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
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, May 05 2020
STATUS
approved