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%I #7 Jul 28 2024 17:08:19
%S 16,18,20,20,24,30,30,36,48,70,90,180,180,420,528,870,1170,2610
%N Areas of trapezoids with integer sides and height whose area equals their perimeter.
%C A trapezoid is a quadrilateral with at least one pair of parallel sides.
%C Conjecture: in this sequence are only three terms which belong to trapezoids with exactly one pair of parallel sides: a(3) = 20, a(5) = 24, a(7) = 30.
%H Felix Huber, <a href="/A374594/a374594.pdf">Illustration of terms a(1) to a(10)</a>.
%H Felix Huber, <a href="/A374594/a374594_1.pdf">Sides and heights of the trapezoids belonging to the terms a(1) to a(18)</a>.
%H Eric Weisstein's World of Mathematics,<a href="https://mathworld.wolfram.com/Trapezoid.html">Trapezoid</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Trapezoid">Trapezoid</a>.
%e See attached illustration of the terms a(1) to a(10).
%p with(NumberTheory):
%p A374594:=proc(k);
%p local K,L,S,T,i,a,c,x,y,h,b,d;
%p L := Divisors(k);
%p S:=[];
%p T:=[];
%p K:=[];
%p for i to numelems(L) do
%p for c to L[i] do
%p a:=2*L[i]-c;
%p h:=k/L[i];
%p x:=0;
%p while x^2<(k-a-c)^2-h^2 do
%p if issqr(x^2+h^2) then
%p d:=sqrt(x^2+h^2);
%p b:=k-a-c-d;
%p y:=a-c-x;
%p if h^2+y^2=b^2 then
%p S:=[a,b,c,d];
%p S:=sort(S);
%p if member(S,T)=false then
%p T:=[op(T),S];
%p K:=[op(K),k];
%p fi;
%p fi;
%p fi;
%p x:=x+1;
%p od;
%p od;
%p od;
%p if numelems(K)>0 then
%p return op(K)
%p fi;
%p end proc;
%p seq(A374594(k),k=1..3000);
%Y Cf. A098030, A181945, A189415, A214602, A272459, A335187, A340858, A348143, A360790.
%K nonn,more
%O 1,1
%A _Felix Huber_, Jul 13 2024
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