A378148 -id:A378148 - cp's OEIS frontend

A378149 a(n) is the number of distinct integer-sided right trapezoids with exactly one pair of parallel sides and area n.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 3, 0, 1, 1, 1, 1, 2, 0, 1, 1, 2, 0, 4, 0, 0, 2, 1, 0, 3, 0, 2, 1, 0, 0, 4, 1, 1, 1, 1, 0, 3, 0, 1, 2, 1, 1, 5, 0, 1, 1, 2, 0, 4, 0, 1, 3, 1, 0, 5, 0, 2, 2, 1, 0, 3, 1, 1, 1

Offset: 1

Felix Huber, Dec 04 2024

nonn

			a(54) = 4 because there are 4 distinct integer-sided right trapezoids [p, r, q, d, h] (p and q are parallel, r is rectangular to p and q, height h = r) with area 54: [13, 6, 5, 10, 6], [20, 3, 16, 5, 3], [7, 12, 2, 13, 12], [15, 4, 12, 5, 4].

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Integer-sided right trapezoids with area n
Eric Weisstein's World of Mathematics, Right Trapezoid

Cf. A027750, A214602, A340858, A340859, A340860, A374594, A378148, A378150.

A378149:=proc(n)
   local a,m,q,M;
   a:=0;
   M:=map(x->x/2, NumberTheory:-Divisors(2*n) minus {1, 2});
   for m in M do
      for q from 1 to m-3/2 do
         if issqr((2*(m-q))^2+(n/m)^2) then
            a:=a+1
         fi
      od
   od;
   return a
end proc;
seq(A378149(n),n=1..87);

a(p) = 0 for prime p.

A378150 a(n) is the number of distinct integer-sided isosceles trapezoids with exactly one pair of parallel sides and area n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 2, 1, 0, 0, 3, 0, 0, 1, 2, 1, 2, 0, 1, 1, 1, 0, 5, 0, 0, 2, 1, 0, 2, 0, 3, 1, 0, 0, 4, 1, 0, 1, 2

Offset: 1

Felix Huber, Dec 02 2024

nonn

Integer-sided isosceles trapezoids with integer area have an integer height. Proof: In an isosceles trapezoid with integer sides and parallel sides p, q with p = q + 2*x, the denominator of x must not be greater than 2. Let us consider the right-angled triangle x, h, d: Assuming that h is not an integer, then x cannot be an integer either, since x = sqrt(d^2 - h^2). Therefore x = (2*s - 1)/2 where s is a positive integer. Since h = 2*n/(p + q) is rational and h = sqrt(d^2 - x^2), it follows that h = (2*t - 1)/2 where t is a positive integer and d^2 = s^2 - s + t^2 - t + 1/2. d is therefore not an integer. It follows that isosceles trapezoids with integer sides and area also have an integer height.

			a(54) = 2 because there are 2 distinct integer-sided isosceles trapezoids [p, d, q, d, h] (p and q are parallel, height h) with area 54: [17, 10, 1, 10, 6], [22, 5, 14, 5, 3].
See also linked Maple program "Integer-sided isosceles trapezoids with area n".

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Integer-sided isosceles trapezoids with area n
Eric Weisstein's World of Mathematics, Isosceles Trapezoid

Cf. A027750, A214602, A340858, A340859, A340860, A374594, A378148, A378149.

A378150:=proc(n)
   local a,m,q,M;
   a:=0;
   M:=NumberTheory:-Divisors(n) minus {1};
   for m in M do
      for q from 1 to m-3 do
         if issqr(((m-q))^2+(n/m)^2) then
            a:=a+1;
         fi
      od
   od;
   return a
end proc;
seq(A378150(n),n=1..88);

a(p) = 0 for prime p.

A378675 Areas of trapezoids with exactly one pair of parallel sides having prime sides and height.

Original entry on oeis.org

15, 21, 27, 27, 45, 45, 55, 63, 65, 81, 85, 85, 95, 99, 115, 117, 125, 125, 135, 145, 155, 171, 175, 175, 185, 189, 205, 207, 225, 235, 243, 245, 265, 275, 279, 295, 297, 315, 315, 325, 333, 335, 355, 365, 385, 387, 405, 407, 425, 451, 455, 459, 473, 475, 475

Offset: 1

Felix Huber, Dec 04 2024

nonn

			27 is twice in the sequence because there are two distinct trapezoids [p, d, q, f, h] (p and q are parallel, height h) with prime sides and height and area 27: [13, 5, 5, 5, 3], [11, 3, 7, 5, 3].

Felix Huber, Table of n, a(n) for n = 1..2633
Felix Huber, Trapezoids having prime sides and height with area A

Cf. A000040, A027750, A070088, A214602, A365049, A366398, A374594, A378148, A378149, A378150.

with(NumberTheory):
A378675:=proc(A)
   local m,p,q,i,j,d,f,h,x,y,M,T;
   if isprime(A)=false and A>1 then
      T:=[];
      M:=map(x->A/x,select(isprime,(Divisors(A)) minus {2}));
      for m in M do
         for i to pi(floor(m-1/2)) do
            q:=ithprime(i);
            p:=2*m-q;
            if isprime(p) then
               h:=A/m;
	       for x from max(4,floor((p-q+1)/2)) by 2 to (h^2-1)/2 do
	          y:=p-q-x;
	          if issqr(x^2+h^2) and issqr(y^2+h^2) then
	             d:=isqrt(y^2+h^2);
	             f:=isqrt(x^2+h^2);
	             if isprime(d) and isprime(f) then
	                T:=[op(T),A]
	             fi
	          fi
	       od
	    fi
         od
      od;
      return op(T)
   fi;
end proc;
seq(A378675(A),A=1..475);

cp's OEIS Frontend

A378149 a(n) is the number of distinct integer-sided right trapezoids with exactly one pair of parallel sides and area n.

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A378150 a(n) is the number of distinct integer-sided isosceles trapezoids with exactly one pair of parallel sides and area n.

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Keywords

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Examples

Links

Crossrefs

Programs

Maple

Formula

A378675 Areas of trapezoids with exactly one pair of parallel sides having prime sides and height.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple