login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A307666
Number of partitions of n into consecutive positive triangular numbers.
5
1, 0, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2
OFFSET
1,10
COMMENTS
Equivalently, number of ways n can be expressed as the difference between two tetrahedral numbers. - Charlie Neder, Apr 24 2019
Records: a(10)=2, a(2180)=3, a(10053736)=4. - Robert Israel, Aug 20 2019
LINKS
FORMULA
G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k*(k+1)/2).
EXAMPLE
10 = 1 + 3 + 6, so a(10) = 2.
MAPLE
N:= 100:
V:= Vector(N):
for i from 1 while i*(i+1)/2 <= N do
s:= i*(i+1)*(i+2)/6;
for j from i-1 to 0 by -1 do
t:= j*(j+1)*(j+2)/6;
if s-t > N then break fi;
V[s-t]:= V[s-t]+1
od;
od:
convert(V, list); # Robert Israel, Aug 20 2019
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 20 2019
STATUS
approved