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

%I A307666 #21 Aug 21 2019 03:15:57
%S A307666 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 A307666 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 A307666 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 A307666 Number of partitions of n into consecutive positive triangular numbers.
%C A307666 Equivalently, number of ways n can be expressed as the difference between two tetrahedral numbers. - _Charlie Neder_, Apr 24 2019
%C A307666 Records: a(10)=2, a(2180)=3, a(10053736)=4. - _Robert Israel_, Aug 20 2019
%H A307666 Robert Israel, <a href="/A307666/b307666.txt">Table of n, a(n) for n = 1..10000</a>
%F A307666 G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k*(k+1)/2).
%e A307666 10 = 1 + 3 + 6, so a(10) = 2.
%p A307666 N:= 100:
%p A307666 V:= Vector(N):
%p A307666 for i from 1 while i*(i+1)/2 <= N do
%p A307666   s:= i*(i+1)*(i+2)/6;
%p A307666   for j from i-1 to 0 by -1 do
%p A307666     t:= j*(j+1)*(j+2)/6;
%p A307666     if s-t > N then break fi;
%p A307666     V[s-t]:= V[s-t]+1
%p A307666   od;
%p A307666 od:
%p A307666 convert(V,list); # _Robert Israel_, Aug 20 2019
%Y A307666 Cf. A000217, A001227, A007294, A024940, A034706, A296338, A307614, A309783.
%K A307666 nonn
%O A307666 1,10
%A A307666 _Ilya Gutkovskiy_, Apr 20 2019