A307666 Number of partitions of n into consecutive positive triangular numbers.

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

Author

Ilya Gutkovskiy, Apr 20 2019

Keywords

• nonn

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

Examples

			10 = 1 + 3 + 6, so a(10) = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A001227, A007294, A024940, A034706, A296338, A307614, A309783.

Programs

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

Formula

G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k*(k+1)/2).