cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A378148 a(n) is the number of distinct trapezoids having integer sides and height with exactly one pair of parallel sides and area n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 3, 0, 1, 2, 1, 0, 3, 0, 1, 2, 1, 0, 4, 0, 2, 2, 1, 1, 5, 0, 1, 2, 3, 0, 5, 0, 2, 3, 1, 0, 6, 0, 2, 2, 2, 0, 7, 1, 3, 2, 1, 0, 9, 0, 1, 3, 3, 2, 8, 0, 3, 2, 3, 0, 10, 0, 1, 5, 3, 0, 9, 0, 6, 3, 1, 0, 10, 2, 1, 2
Offset: 1

Views

Author

Felix Huber, Dec 02 2024

Keywords

Comments

The number of trapezoids having integer sides and height, which are neither right-angled nor isosceles, is a(n) - A378149(n) - A378150(n). The first trapezoid, which is neither right-angled nor isosceles, appears at a(36).
a(p) = 0 for prime p. Proof: Suppose there is a trapezoid with integer sides and prime area p. Then in p = m*h (m is the average of the parallel sides and h is the height of the trapezoid) m = p and h = 1 or m = p/2 and h = 2. At least one nonparallel side of the trapezoid is the hypotenuse of a right triangle with leg h. Legs in integer right triangles are >= 3. This is a contradiction and therefore a(p) = 0.
A214602 is the index of the positive terms in this sequence.
There are also integer-sided trapezoids with integer area that do not have an integer height. For example, the trapezoid with sides p = 630, d = 615, q = 5, f = 40 (p and q are parallel) has an area of 12192 and a height of h = 38.4.

Examples

			a(54) = 7 because there are 7 distinct trapezoids [p, d, q, f, h] (p and q are parallel, height h) having integer sides and height with area 54:[17, 10, 1, 10, 6], [13, 6, 5, 10, 6], [22, 5, 14, 5, 3], [20, 3, 16, 5, 3], [8, 15, 1, 20, 12], [7, 12, 2, 13, 12], [15, 4, 12, 5, 4].
For a(54) = 7 and (92) = 4 see the linked illustrations.
See also the linked Maple program "Trapezoids having integer sides and height with area n".

Links

Crossrefs

Cf. A024406, A027750, A103606, A214602, A340858, A340859, A340860, A365049, A374594, A378149, A378150.

Programs

  • Maple
    A378148:=proc(n)
   local a,m,p,q,h,x,y,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-1/2 do
         p:=2*m-q;
         h:=n/m;
         for x from max(3,floor((p-q+1)/2)) to (h^2-1)/2 do
            y:=p-q-x;
            if issqr(x^2+h^2) and issqr(y^2+h^2) then
               a:=a+1
            fi
         od
      od
   od;
   return a
end proc;
seq(A378148(n),n=1..87);

Formula

a(p) = 0 for prime p.

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

Original entry on oeis.org

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

Views

Author

Felix Huber, Dec 04 2024

Keywords

Examples

			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].

Links

Crossrefs

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

Programs

  • Maple
    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);

Formula

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

Views

Author

Felix Huber, Dec 02 2024

Keywords

Comments

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.

Examples

			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".

Links

Crossrefs

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

Programs

  • Maple
    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);

Formula

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

Views

Author

Felix Huber, Dec 04 2024

Keywords

Examples

			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].

Links

Crossrefs

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

Programs

  • Maple
    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);
Showing 1-4 of 4 results.

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