%I #21 Aug 21 2019 03:15:57

%S 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,

%T 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,

%U 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

%N Number of partitions of n into consecutive positive triangular numbers.

%C Equivalently, number of ways n can be expressed as the difference between two tetrahedral numbers. - _Charlie Neder_, Apr 24 2019

%C Records: a(10)=2, a(2180)=3, a(10053736)=4. - _Robert Israel_, Aug 20 2019

%H Robert Israel, <a href="/A307666/b307666.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%p N:= 100:

%p V:= Vector(N):

%p for i from 1 while i*(i+1)/2 <= N do

%p s:= i*(i+1)*(i+2)/6;

%p for j from i-1 to 0 by -1 do

%p t:= j*(j+1)*(j+2)/6;

%p if s-t > N then break fi;

%p V[s-t]:= V[s-t]+1

%p od;

%p od:

%p convert(V,list); # _Robert Israel_, Aug 20 2019

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

%K nonn

%O 1,10

%A _Ilya Gutkovskiy_, Apr 20 2019