A378961 Number of sets of nonzero triangular numbers whose largest element is the n-th triangular number and whose sum is a triangular number.

1, 1, 2, 1, 3, 5, 5, 11, 19, 33, 55, 92, 192, 327, 579, 1142, 2052, 3776, 6936, 12964, 24308, 44432, 84763, 159299, 299093, 567295, 1075570, 2045580, 3883453, 7411014, 14164089, 27044407, 51759660, 99259961, 190371661, 365537357, 702901278, 1352868238, 2606296357

Offset: 1

Author

Ilya Gutkovskiy, Dec 12 2024

Keywords

• nonn

Examples

			a(5) = 3 subsets of triangular numbers whose largest element is A000217(5)=15 and whose sum is in A000217: {15}, {6, 15} and {3, 10, 15}.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..201

Crossrefs

Cf. A000217, A339612, A339613, A377123 (partial sums).

Programs

Maple

istri:= proc(n) issqr(1+8*n) end proc:
tri:= n -> n*(n+1)/2:
F:= proc(n,s) option remember; local v;
  if s = 0 then return 1 fi;
  if s > n*(n+1)*(n+2)/6 then return 0 fi;
  v:= tri(n);
  if s >= v then procname(n-1,s-v) + procname(n-1,s)
  else procname(n-1,s)
  fi;
end proc:
f:= proc(n) local i,t,m;
   t:= 0;
   m:= n*(n+1)*(n+2)/6;
   for i from 1 while tri(i) <= m do
     t:= t + F(n,tri(i)) - F(n-1,tri(i))
   od;
   t
end proc:
map(f, [$1..50]); # Robert Israel, Jan 13 2025