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

%I A378149 #10 Dec 23 2024 22:18:49
%S A378149 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,
%T A378149 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,
%U A378149 1,2,0,4,0,1,3,1,0,5,0,2,2,1,0,3,1,1,1
%N A378149 a(n) is the number of distinct integer-sided right trapezoids with exactly one pair of parallel sides and area n.
%H A378149 Felix Huber, <a href="/A378149/b378149.txt">Table of n, a(n) for n = 1..10000</a>
%H A378149 Felix Huber, <a href="/A378149/a378149.txt">Integer-sided right trapezoids with area n</a>
%H A378149 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RightTrapezoid.html">Right Trapezoid</a>
%F A378149 a(p) = 0 for prime p.
%e A378149 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].
%p A378149 A378149:=proc(n)
%p A378149    local a,m,q,M;
%p A378149    a:=0;
%p A378149    M:=map(x->x/2, NumberTheory:-Divisors(2*n) minus {1, 2});
%p A378149    for m in M do
%p A378149       for q from 1 to m-3/2 do
%p A378149          if issqr((2*(m-q))^2+(n/m)^2) then
%p A378149             a:=a+1
%p A378149          fi
%p A378149       od
%p A378149    od;
%p A378149    return a
%p A378149 end proc;
%p A378149 seq(A378149(n),n=1..87);
%Y A378149 Cf. A027750, A214602, A340858, A340859, A340860, A374594, A378148, A378150.
%K A378149 nonn
%O A378149 1,18
%A A378149 _Felix Huber_, Dec 04 2024