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

16, 18, 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 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.

Links

Table of n, a(n) for n=1..19.

Felix Huber, Illustration of terms a(1) to a(11)

Felix Huber, Sides and heights of the trapezoids belonging to the terms a(1) to a(19)

Eric Weisstein's World of Mathematics,Trapezoid.

Wikipedia, Trapezoid.

Example

See attached illustration of the terms a(1) to a(11).

Maple

with(NumberTheory):
A374594:=proc(k);
  local K, L, S, T, i, a, c, x, y, h, b, d;
  L:=map(x->x/2, Divisors(2*k) minus {1, 2});
  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:=isqrt(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: A002925 A070540 A341271 * A358677 A085096 A043675

Adjacent sequences: A374591 A374592 A374593 * A374595 A374596 A374597

Keyword

nonn,more

Author

Felix Huber, Jul 13 2024

Extensions

Corrected by Felix Huber, Dec 04 2024

Status

approved