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A374594 Areas of trapezoids with integer sides and height whose area equals their perimeter.

%I A374594 #11 Dec 17 2024 17:25:31
%S A374594 16,18,18,20,20,24,30,30,36,48,70,90,180,180,420,528,870,1170,2610
%N A374594 Areas of trapezoids with integer sides and height whose area equals their perimeter.
%C A374594 A trapezoid is a quadrilateral with at least one pair of parallel sides.
%C A374594 Conjecture: in this sequence are only four terms which belong to trapezoids with exactly one pair of parallel sides: a(2) = 18, a(4) = 20, a(6) = 24, a(7) = 30.
%H A374594 Felix Huber, <a href="/A374594/a374594_2.pdf">Illustration of terms a(1) to a(11)</a>
%H A374594 Felix Huber, <a href="/A374594/a374594_3.pdf">Sides and heights of the trapezoids belonging to the terms a(1) to a(19)</a>
%H A374594 Eric Weisstein's World of Mathematics,<a href="https://mathworld.wolfram.com/Trapezoid.html">Trapezoid</a>.
%H A374594 Wikipedia, <a href="https://en.wikipedia.org/wiki/Trapezoid">Trapezoid</a>.
%e A374594 See attached illustration of the terms a(1) to a(11).
%p A374594 with(NumberTheory):
%p A374594 A374594:=proc(k);
%p A374594   local K,L,S,T,i,a,c,x,y,h,b,d;
%p A374594   L:=map(x->x/2, Divisors(2*k) minus {1, 2});
%p A374594   S:=[];
%p A374594   T:=[];
%p A374594   K:=[];
%p A374594   for i to numelems(L) do
%p A374594     for c to L[i] do
%p A374594       a:=2*L[i]-c;
%p A374594       h:=k/L[i];
%p A374594       x:=0;
%p A374594       while x^2<(k-a-c)^2-h^2 do
%p A374594         if issqr(x^2+h^2) then
%p A374594           d:=isqrt(x^2+h^2);
%p A374594           b:=k-a-c-d;
%p A374594           y:=a-c-x;
%p A374594           if h^2+y^2=b^2 then
%p A374594             S:=[a,b,c,d];
%p A374594             S:=sort(S);
%p A374594             if member(S,T)=false then
%p A374594               T:=[op(T),S];
%p A374594               K:=[op(K),k];
%p A374594             fi;
%p A374594           fi;
%p A374594         fi;
%p A374594         x:=x+1;
%p A374594       od;
%p A374594     od;
%p A374594   od;
%p A374594   if numelems(K)>0 then
%p A374594     return op(K)
%p A374594   fi;
%p A374594 end proc;
%p A374594 seq(A374594(k),k=1..3000);
%Y A374594 Cf. A098030, A181945, A189415, A214602, A272459, A335187, A340858, A348143, A360790.
%K A374594 nonn,more
%O A374594 1,1
%A A374594 _Felix Huber_, Jul 13 2024
%E A374594 Corrected by _Felix Huber_, Dec 04 2024