A378150 a(n) is the number of distinct integer-sided isosceles trapezoids with exactly one pair of parallel sides and area n.

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

Offset

1,24

Comments

Integer-sided isosceles trapezoids with integer area have an integer height. Proof: In an isosceles trapezoid with integer sides and parallel sides p, q with p = q + 2*x, the denominator of x must not be greater than 2. Let us consider the right-angled triangle x, h, d: Assuming that h is not an integer, then x cannot be an integer either, since x = sqrt(d^2 - h^2). Therefore x = (2*s - 1)/2 where s is a positive integer. Since h = 2*n/(p + q) is rational and h = sqrt(d^2 - x^2), it follows that h = (2*t - 1)/2 where t is a positive integer and d^2 = s^2 - s + t^2 - t + 1/2. d is therefore not an integer. It follows that isosceles trapezoids with integer sides and area also have an integer height.

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Felix Huber, Integer-sided isosceles trapezoids with area n

Eric Weisstein's World of Mathematics, Isosceles Trapezoid

Formula

a(p) = 0 for prime p.

Example

a(54) = 2 because there are 2 distinct integer-sided isosceles trapezoids [p, d, q, d, h] (p and q are parallel, height h) with area 54: [17, 10, 1, 10, 6], [22, 5, 14, 5, 3].
See also linked Maple program "Integer-sided isosceles trapezoids with area n".

Maple

A378150:=proc(n)
   local a, m, q, M;
   a:=0;
   M:=NumberTheory:-Divisors(n) minus {1};
   for m in M do
      for q from 1 to m-3 do
         if issqr(((m-q))^2+(n/m)^2) then
            a:=a+1;
         fi
      od
   od;
   return a
end proc;
seq(A378150(n), n=1..88);

Crossrefs

Cf. A027750, A214602, A340858, A340859, A340860, A374594, A378148, A378149.

Sequence in context: A257469 A275947 A337976 * A125005 A378149 A309367

Adjacent sequences: A378147 A378148 A378149 * A378151 A378152 A378153

Keyword

nonn

Author

Felix Huber, Dec 02 2024

Status

approved