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

%I A378150 #8 Dec 03 2024 12:50:40
%S A378150 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,
%T A378150 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,
%U A378150 1,1,0,5,0,0,2,1,0,2,0,3,1,0,0,4,1,0,1,2
%N A378150 a(n) is the number of distinct integer-sided isosceles trapezoids with exactly one pair of parallel sides and area n.
%C A378150 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.
%H A378150 Felix Huber, <a href="/A378150/b378150.txt">Table of n, a(n) for n = 1..10000</a>
%H A378150 Felix Huber, <a href="/A378150/a378150.txt">Integer-sided isosceles trapezoids with area n</a>
%H A378150 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IsoscelesTrapezoid.html">Isosceles Trapezoid</a>
%F A378150 a(p) = 0 for prime p.
%e A378150 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].
%e A378150 See also linked Maple program "Integer-sided isosceles trapezoids with area n".
%p A378150 A378150:=proc(n)
%p A378150    local a,m,q,M;
%p A378150    a:=0;
%p A378150    M:=NumberTheory:-Divisors(n) minus {1};
%p A378150    for m in M do
%p A378150       for q from 1 to m-3 do
%p A378150          if issqr(((m-q))^2+(n/m)^2) then
%p A378150             a:=a+1;
%p A378150          fi
%p A378150       od
%p A378150    od;
%p A378150    return a
%p A378150 end proc;
%p A378150 seq(A378150(n),n=1..88);
%Y A378150 Cf. A027750, A214602, A340858, A340859, A340860, A374594, A378148, A378149.
%K A378150 nonn
%O A378150 1,24
%A A378150 _Felix Huber_, Dec 02 2024