A378149 a(n) is the number of distinct integer-sided right trapezoids with exactly one pair of parallel sides and area n.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 4, 0, 0, 2, 1, 0, 3, 0, 2, 1, 0, 0, 4, 1, 1, 1, 1, 0, 3, 0, 1, 2, 1, 1, 5, 0, 1, 1, 2, 0, 4, 0, 1, 3, 1, 0, 5, 0, 2, 2, 1, 0, 3, 1, 1, 1

Offset: 1

Author

Felix Huber, Dec 04 2024

Keywords

• nonn

Examples

			a(54) = 4 because there are 4 distinct integer-sided right trapezoids [p, r, q, d, h] (p and q are parallel, r is rectangular to p and q, height h = r) with area 54: [13, 6, 5, 10, 6], [20, 3, 16, 5, 3], [7, 12, 2, 13, 12], [15, 4, 12, 5, 4].
		

Links

• Felix Huber, Table of n, a(n) for n = 1..10000
• Felix Huber, Integer-sided right trapezoids with area n
• Eric Weisstein's World of Mathematics, Right Trapezoid

Crossrefs

Cf. A027750, A214602, A340858, A340859, A340860, A374594, A378148, A378150.

Programs

Maple

A378149:=proc(n)
   local a,m,q,M;
   a:=0;
   M:=map(x->x/2, NumberTheory:-Divisors(2*n) minus {1, 2});
   for m in M do
      for q from 1 to m-3/2 do
         if issqr((2*(m-q))^2+(n/m)^2) then
            a:=a+1
         fi
      od
   od;
   return a
end proc;
seq(A378149(n),n=1..87);

Formula

a(p) = 0 for prime p.