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

16, 18, 20, 20, 24, 30, 30, 36, 48, 70, 90, 180, 180, 420, 528, 870, 1170, 2610

Offset

1,1

Comments

A trapezoid is a quadrilateral with at least one pair of parallel sides.

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.

Links

Table of n, a(n) for n=1..18.

Felix Huber, Illustration of terms a(1) to a(10).

Felix Huber, Sides and heights of the trapezoids belonging to the terms a(1) to a(18).

Eric Weisstein's World of Mathematics,Trapezoid.

Wikipedia, Trapezoid.

Example

See attached illustration of the terms a(1) to a(10).

Maple

with(NumberTheory):
A374594:=proc(k);
  local K, L, S, T, i, a, c, x, y, h, b, d;
  L := Divisors(k);
  S:=[];
  T:=[];
  K:=[];
  for i to numelems(L) do
    for c to L[i] do
      a:=2*L[i]-c;
      h:=k/L[i];
      x:=0;
      while x^2<(k-a-c)^2-h^2 do
        if issqr(x^2+h^2) then
          d:=sqrt(x^2+h^2);
          b:=k-a-c-d;
          y:=a-c-x;
          if h^2+y^2=b^2 then
            S:=[a, b, c, d];
            S:=sort(S);
            if member(S, T)=false then
              T:=[op(T), S];
              K:=[op(K), k];
            fi;
          fi;
        fi;
        x:=x+1;
      od;
    od;
  od;
  if numelems(K)>0 then
    return op(K)
  fi;
end proc;
seq(A374594(k), k=1..3000);

Crossrefs

Cf. A098030, A181945, A189415, A214602, A272459, A335187, A340858, A348143, A360790.

Sequence in context: A085096 A043675 A043705 * A178980 A031315 A077914

Adjacent sequences: A374591 A374592 A374593 * A374595 A374596 A374598

Keyword

nonn,more

Author

Felix Huber, Jul 13 2024

Status

approved