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A378148 a(n) is the number of distinct trapezoids having integer sides and height with exactly one pair of parallel sides and area n.

%I A378148 #22 Dec 03 2024 12:50:30
%S A378148 0,0,0,0,0,0,0,0,1,1,0,1,0,1,2,1,0,3,0,1,2,1,0,3,0,1,2,1,0,4,0,2,2,1,
%T A378148 1,5,0,1,2,3,0,5,0,2,3,1,0,6,0,2,2,2,0,7,1,3,2,1,0,9,0,1,3,3,2,8,0,3,
%U A378148 2,3,0,10,0,1,5,3,0,9,0,6,3,1,0,10,2,1,2
%N A378148 a(n) is the number of distinct trapezoids having integer sides and height with exactly one pair of parallel sides and area n.
%C A378148 The number of trapezoids having integer sides and height, which are neither right-angled nor isosceles, is a(n) - A378149(n) - A378150(n). The first trapezoid, which is neither right-angled nor isosceles, appears at a(36).
%C A378148 a(p) = 0 for prime p. Proof: Suppose there is a trapezoid with integer sides and prime area p. Then in p = m*h (m is the average of the parallel sides and h is the height of the trapezoid) m = p and h = 1 or m = p/2 and h = 2. At least one nonparallel side of the trapezoid is the hypotenuse of a right triangle with leg h. Legs in integer right triangles are >= 3. This is a contradiction and therefore a(p) = 0.
%C A378148 A214602 is the index of the positive terms in this sequence.
%C A378148 There are also integer-sided trapezoids with integer area that do not have an integer height. For example, the trapezoid with sides p = 630, d = 615, q = 5, f = 40 (p and q are parallel) has an area of 12192 and a height of h = 38.4.
%H A378148 Felix Huber, <a href="/A378148/b378148.txt">Table of n, a(n) for n = 1..10000</a>
%H A378148 Felix Huber, <a href="/A378148/a378148_1.pdf">Illustration of a(54) = 7</a>
%H A378148 Felix Huber, <a href="/A378148/a378148.pdf">Illustration of a(92) = 4</a>
%H A378148 Felix Huber, <a href="/A378148/a378148_2.txt">Trapezoids having integer sides and height with area n</a>
%H A378148 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Trapezoid.html">Trapezoid</a>
%F A378148 a(p) = 0 for prime p.
%e A378148 a(54) = 7 because there are 7 distinct trapezoids [p, d, q, f, h] (p and q are parallel, height h) having integer sides and height with area 54:[17, 10, 1, 10, 6], [13, 6, 5, 10, 6], [22, 5, 14, 5, 3], [20, 3, 16, 5, 3], [8, 15, 1, 20, 12], [7, 12, 2, 13, 12], [15, 4, 12, 5, 4].
%e A378148 For a(54) = 7 and (92) = 4 see the linked illustrations.
%e A378148 See also the linked Maple program "Trapezoids having integer sides and height with area n".
%p A378148 A378148:=proc(n)
%p A378148    local a,m,p,q,h,x,y,M;
%p A378148    a:=0;
%p A378148    M:=map(x->x/2,NumberTheory:-Divisors(2*n) minus {1,2});
%p A378148    for m in M do
%p A378148       for q from 1 to m-1/2 do
%p A378148          p:=2*m-q;
%p A378148          h:=n/m;
%p A378148          for x from max(3,floor((p-q+1)/2)) to (h^2-1)/2 do
%p A378148             y:=p-q-x;
%p A378148             if issqr(x^2+h^2) and issqr(y^2+h^2) then
%p A378148                a:=a+1
%p A378148             fi
%p A378148          od
%p A378148       od
%p A378148    od;
%p A378148    return a
%p A378148 end proc;
%p A378148 seq(A378148(n),n=1..87);
%Y A378148 Cf. A024406, A027750, A103606, A214602, A340858, A340859, A340860, A365049, A374594, A378149, A378150.
%K A378148 nonn
%O A378148 1,15
%A A378148 _Felix Huber_, Dec 02 2024