A340858 -id:A340858 - cp's OEIS frontend

A342720 a(n) is the number of concave integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d) and integer diagonals e,f.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 3, 1, 2, 2, 3, 1, 4, 2, 4, 2, 5, 3, 7, 1, 2, 4, 3, 13, 7, 20, 12, 5, 3, 7, 10, 3, 8, 2, 14, 12, 10, 15, 17, 8, 11, 10, 20, 13, 15, 10, 45, 9, 18, 25, 46, 38, 18, 2, 25, 20, 30, 18, 32, 17, 32, 43

Offset: 1

Herbert Kociemba, Mar 19 2021

nonn

Without loss of generality we assume that a is the largest side length and that the diagonal e divides the concave quadrilateral into two triangles with sides a,b,e and c,d,e. Then e < a is a necessary condition for concavity. The triangle inequality further implies e > a-b and abs(e-c) < d < e+c.

			a(15)=1 because the only concave integer quadrilateral with longest edge length 15 has a=15, b=13, c=13, d=15 and diagonals e=4 and f=24. a(20)=3 because there are three solutions (a,b,c,d,e,f): (20,13,15,18,9,26), (20,13,13,20,11,24) and {20,15,15,20,7,24}.

Cf. A340858 for trapezoids, A342721 for concave integer quadrilaterals with integer area.

an={};
he[a_,b_,e_]:=1/(2 e) Sqrt[(-((a-b-e) (a+b-e) (a-b+e) (a+b+e)))]
paX[e_]:={e,0} (*vertex A coordinate*)
pbX[a_,b_,e_]:={(-a^2+b^2+e^2)/(2 e),he[a,b,e]}(*vertex B coordinate*)
pc={0,0};(*vertex C coordinate*)
pdX[c_,d_,e_]:={(c^2-d^2+e^2)/(2 e),-he[c,d,e]}(*vertex D coordinate*)
concaveQ[{bx_,by_},{dx_,dy_},e_]:=If[by dx-bx dy<0||by dx-bx dy>(by-dy) e,True,False]
gQ[x_,y_]:=Module[{z=x-y,res=False},Do[If[z[[i]]>0,res=True;Break[],
  If[z[[i]]<0,Break[]]],{i,1,4}];res]
canonicalQ[{a_,b_,c_,d_}]:=Module[{m={a,b,c,d}},If[(gQ[{b,a,d,c},m]||gQ[{d,c,b,a},m]||gQ[{c,d,a,b},m]),False,True]]
Do[cnt=0;
Do[pa=paX[e];pb=pbX[a,b,e];pd=pdX[c,d,e];
If[(f=Sqrt[(pb-pd).(pb-pd)];IntegerQ[f])&&concaveQ[pb,pd,e]&&canonicalQ[{a,b,c,d}],cnt++
(*;Print[{{a,b,c,d,e,f},Graphics[Line[{pa,pb,pc,pd,pa}]]}]*)],
{b,1,a},{e,a-b+1,a-1},{c,1,a},{d,Abs[e-c]+1,Min[a,e+c-1]}];
AppendTo[an,cnt],
{a,1,75}
]
an

A340859 a(n) is the number of isosceles integer trapezoids (up to congruence) with integer side lengths a,c,b=d with n=Max(a,b,c) and integer diagonals e=f.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 5, 6, 3, 3, 9, 6, 5, 10, 20, 9, 10, 8, 21, 18, 10, 10, 37, 21, 12, 24, 31, 14, 26, 17, 55, 32, 20, 36, 54, 22, 20, 39, 74, 24, 40, 26, 58, 59, 24, 26, 113, 47, 41, 54, 69, 33, 51, 61, 111, 65, 35, 39, 124, 38, 39, 88, 145, 79

Offset: 1

Herbert Kociemba, Jan 24 2021

nonn

By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.

Without loss of generality we assume b=d and for the parallel sides c < a. e and f are uniquely determined by e = f = sqrt((c(a^2-b^2) + a(b^2-c^2))/(a-c)). The smallest possible isosceles trapezoid has side lengths a=4, c=3, b=d=2 and diagonals e=f=4.

			a(7)=2 because there are two possible trapezoids: a=5, c=3, b=d=7, e=f=8 and a=7, c=4, b=d=6, e=f=8.

Cf. A224931 for parallelograms, A340858 for general trapezoids and A340860 for non-isosceles trapezoids.

n=65;list={};
For[a=1,a<=n,a++,
For[c=1,cse,Break[]];If[sf<=0,Continue[]];
e=Sqrt[se/(a-c)];f=Sqrt[sf/(a-c)];
If[IntegerQ[e]&&IntegerQ[f]&&a+d>f&&d+f>a&&f+a>d&&e+b>a&&b+a>e&&a+e>b,AppendTo[list,{a,b,c,d,e,f}]]]]]]
Table[Select[list,Max[#[[1]],#[[2]],#[[3]],#[[4]]]==n&&#[[2]]==#[[4]]&]//Length,{n,1,65}]

A340860 a(n) is the number of non-isosceles integer trapezoids (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d) and integer diagonals e,f.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 2, 0, 2, 1, 0, 1, 1, 0, 4, 4, 9, 5, 9, 11, 7, 4, 5, 3, 11, 13, 2, 7, 3, 3, 5, 9, 8, 3, 6, 9, 12, 10, 19, 8, 23, 16, 16, 18, 21, 13, 25, 19, 32, 26, 7, 7, 25, 16, 8, 27, 59, 26

Offset: 1

Herbert Kociemba, Jan 24 2021

nonn

By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.
Without loss of generality we assume for the parallel sides c < a and for the diagonals f < e. e and f are uniquely determined by e = sqrt((c(a^2-b^2) + a(d^2-c^2))/(a-c)) and f = sqrt((c(a^2-d^2) + a(b^2-c^2))/(a-c)).
The smallest possible trapezoid which is not isosceles has side lengths a=8, b=9, c=3, d=11 and diagonals e=13 and f=9.

			a(34)=2 because up to congruence there are exactly two trapezoids which are not isosceles:
a=32, b=26, c=22, d=34 and e=54, f=18;
a=34, b=11, c=32, d=12 and e=40, f=29.

Cf. A224931 for parallelograms, A340858 for general trapezoids and A340859 for isosceles trapezoids.

n=65;list={};
For[a=1,a<=n,a++,
For[c=1,cse,Break[]];If[sf<=0,Continue[]];
e=Sqrt[se/(a-c)];f=Sqrt[sf/(a-c)];
If[IntegerQ[e]&&IntegerQ[f]&&a+d>f&&d+f>a&&f+a>d&&e+b>a&&b+a>e&&a+e>b,AppendTo[list,{a,b,c,d,e,f}]]]]]]
Table[Select[list,Max[#[[1]],#[[2]],#[[3]],#[[4]]]==n&&#[[2]]!=#[[4]]&]//Length,{n,1,65}]

A342721 a(n) is the number of concave integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d), integer diagonals e,f and integer area.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 3, 1, 1, 0, 0, 1, 3, 0, 0, 0, 2, 1, 0, 6, 0, 4, 4, 2, 1, 0, 0, 1, 0, 0, 6, 0, 2, 8, 6, 2, 0, 1, 2, 0, 2, 0, 9, 0, 0, 2, 0, 13, 1, 0, 4, 0, 3, 0, 3, 5, 10, 11

Offset: 1

Herbert Kociemba, Mar 19 2021

nonn

Without loss of generality we assume that a is the largest side length and that the diagonal e divides the concave quadrilateral into two triangles with sides a,b,e and c,d,e. Then e < a is a necessary condition for concavity. The triangle inequality further implies e > a-b and abs(e-c) < d < e+c.

			a(66)=1 because the only concave integer quadrilateral with longest edge length 66 and integer area has sides a=66, b=55, c=12, d=65, diagonals e=55, f=65 and area 1650.

Cf. A340858 for trapezoids, A342720 for concave integer quadrilaterals with arbitrary area.

an={};
area[a_,b_,c_,d_,e_,f_]:=(1/4)Sqrt[(4e^2 f^2-(a^2+c^2-b^2-d^2)^2)]
he[a_,b_,e_]:=(1/(2 e))Sqrt[(-((a-b-e) (a+b-e) (a-b+e) (a+b+e)))];
paX[e_]:={e,0} (*vertex A coordinate*)
pbX[a_,b_,e_]:={(-a^2+b^2+e^2)/(2 e),he[a,b,e]}(*vertex B coordinate*)
pc={0,0};(*vertex C coordinate*)
pdX[c_,d_,e_]:={(c^2-d^2+e^2)/(2 e),-he[c,d,e]}(*vertex D coordinate*)
concaveQ[{bx_,by_},{dx_,dy_},e_]:=If[by dx-bx dy<0||by dx-bx dy>(by-dy) e,True,False]
gQ[x_,y_]:=Module[{z=x-y,res=False},Do[If[z[[i]]>0,res=True;Break[],If[z[[i]]<0,Break[]]],{i,1,4}];res]
canonicalQ[{a_,b_,c_,d_}]:=Module[{m={a,b,c,d}},If[(gQ[{b,a,d,c},m]||gQ[{d,c,b,a},m]||gQ[{c,d,a,b},m]),False,True]]
Do[cnt=0;
Do[pa=paX[e];pb=pbX[a,b,e];pd=pdX[c,d,e];
If[(f=Sqrt[(pb-pd).(pb-pd)];IntegerQ[f])&&(ar=area[a,b,c,d,e,f]; IntegerQ[ar])&&concaveQ[pb,pd,e]&&canonicalQ[{a,b,c,d}],cnt++
(*;Print[{{a,b,c,d,e,f,ar},Graphics[Line[{pa,pb,pc,pd,pa}]]}]*)],
{b,1,a},{e,a-b+1,a-1},{c,1,a},{d,Abs[e-c]+1,Min[a,e+c-1]}];
AppendTo[an,cnt],{a,1,75}
]
an

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

Original entry on oeis.org

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

Offset: 1

Felix Huber, Jul 13 2024

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.

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

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.

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

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);

Corrected by Felix Huber, Dec 04 2024

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

Felix Huber, Dec 02 2024

nonn

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.

			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".

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Illustration of a(54) = 7
Felix Huber, Illustration of a(92) = 4
Felix Huber, Trapezoids having integer sides and height with area n
Eric Weisstein's World of Mathematics, Trapezoid

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

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);

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

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.

A342722 a(n) is the number of convex integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d) and integer diagonals e,f.

Original entry on oeis.org

0, 0, 0, 2, 2, 1, 5, 7, 8, 5, 7, 13, 14, 11, 15, 31, 18, 14, 18, 30, 25, 24, 22, 64, 42, 35, 51, 58, 34, 48, 37, 87, 71, 46, 69, 74, 51, 53, 74, 110, 53, 72, 61, 96, 106, 73, 60, 181, 102, 103, 125, 134, 79, 118, 133, 215, 141, 82, 82, 221

Offset: 1

Herbert Kociemba, Apr 25 2021

nonn

Without loss of generality we assume that a is the largest side length and that the diagonal e divides the convex quadrilateral into two triangles with sides a,b,e and c,d,e. The triangle inequality implies e > a-b and abs(e-c) < d < e+c.

			a(6)=1 because the only convex integer quadrilateral with longest edge length 6 is a trapezoid with sides a=6, b=5, c=4, d=5 and diagonals e=f=7.

Cf. A340858 for trapezoids, A342720 and A342721 for concave integer quadrilaterals, A342723 for convex integer quadrilaterals with integer area.

an={};
he[a_,b_,e_]:=1/(2 e) Sqrt[-(a-b-e) (a+b-e) (a-b+e) (a+b+e)];
paX[e_]:={e,0} (*vertex A coordinate*)
pbX[a_,b_,e_]:={(-a^2+b^2+e^2)/(2 e),he[a,b,e]}(*vertex B coordinate*)
pc={0,0};(*vertex C coordinate*)
pdX[c_,d_,e_]:={(c^2-d^2+e^2)/(2 e),-he[c,d,e]}(*vertex D coordinate*)
convexQ[{bx_,by_},{dx_,dy_},e_]:=If[(by-dy) e>by dx-bx dy>0,True,False]
(*define order on tuples*)
gQ[x_,y_]:=Module[{z=x-y,res=False},Do[If[z[[i]]>0,res=True;Break[],If[z[[i]]<0,Break[]]],{i,1,6}];res]
(*check if tuple is canonical*)
canonicalQ[{a_,b_,c_,d_,e_,f_}]:=Module[{x={a,b,c,d,e,f}},If[(gQ[{b,a,d,c,e,f},x]||gQ[{d,c,b,a,e,f},x]||gQ[{c,d,a,b,e,f},x]||gQ[{b,c,d,a,f,e},x]||gQ[{a,d,c,b,f,e},x]||gQ[{c,b,a,d,f,e},x]||gQ[{d,a,b,c,f,e},x]),False,True]]
Do[cnt=0;
Do[pa=paX[e];pb=pbX[a,b,e];pd=pdX[c,d,e];
If[(f=Sqrt[(pb-pd).(pb-pd)];IntegerQ[f])&&convexQ[pb,pd,e]&&canonicalQ[{a,b,c,d,e,f}],cnt++
(*;Print[{{a,b,c,d,e,f},Graphics[Line[{pa,pb,pc,pd,pa}]]}]*)],
{b,1,a},{e,a-b+1,a+b-1},{c,1,a},{d,Abs[e-c]+1,Min[a,e+c-1]}];
AppendTo[an,cnt],{a,1 ,60}
]
an

A342723 a(n) is the number of convex integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d), integer diagonals e,f and integer area.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 3, 1, 2, 0, 0, 4, 3, 0, 0, 4, 7, 2, 0, 5, 2, 5, 0, 1, 0, 3, 4, 3, 4, 0, 6, 7, 3, 4, 0, 3, 4, 0, 0, 5, 0, 9, 10, 9, 3, 0, 5, 8, 0, 4, 0, 17, 4, 0, 9, 1, 19, 2, 0, 6, 2, 7, 0, 7, 7, 7, 23, 2, 8, 12, 0, 10, 0, 5, 0, 15, 27

Offset: 1

Herbert Kociemba, Apr 25 2021

nonn

Without loss of generality we assume that a is the largest side length and that the diagonal e divides the convex quadrilateral into two triangles with sides a,b,e and c,d,e. The triangle inequality implies e > a-b and abs(e-c) < d < e+c.

			a(4)=1 is the smallest possible solution and is a rectangle with a=c=4, b=d=3, e=f=5 and area 12. a(24)=4 includes the smallest possible solution with all sides a,b,c,d different and a=24, b=20, c=15, d=7, e=20, f=25 and area 234. Furthermore there are three rectangles with a=24,b=7, a=24,b=10 and a=24,b=18.

Cf. A340858 for trapezoids, A342720 and A342721 for concave integer quadrilaterals, A342722 for convex integer quadrilaterals with arbitrary area.

an={};
area[a_,b_,c_,d_,e_,f_]:=1/4 Sqrt[4e^2 f^2-(a^2+c^2-b^2-d^2)^2];
he[a_,b_,e_]:=1/(2 e) Sqrt[-(a-b-e) (a+b-e) (a-b+e) (a+b+e)];
paX[e_]:={e,0} (*vertex A coordinate*)
pbX[a_,b_,e_]:={(-a^2+b^2+e^2)/(2 e),he[a,b,e]}(*vertex B coordinate*)
pc={0,0};(*vertex C coordinate*)pdX[c_,d_,e_]:={(c^2-d^2+e^2)/(2 e),-he[c,d,e]}(*vertex D coordinate*)
convexQ[{bx_,by_},{dx_,dy_},e_]:=If[(by-dy) e>by dx-bx dy>0,True,False]
(*define order on tuples*)
gQ[x_,y_]:=Module[{z=x-y,res=False},Do[If[z[[i]]>0,res=True;Break[],If[z[[i]]<0,Break[]]],{i,1,6}];res]
(*check if tuple is canonical*)
canonicalQ[{a_,b_,c_,d_,e_,f_}]:=Module[{x={a,b,c,d,e,f}},If[(gQ[{b,a,d,c,e,f},x]||gQ[{d,c,b,a,e,f},x]||gQ[{c,d,a,b,e,f},x]||gQ[{b,c,d,a,f,e},x]||gQ[{a,d,c,b,f,e},x]||gQ[{c,b,a,d,f,e},x]||gQ[{d,a,b,c,f,e},x]),False,True]]
Do[cnt=0;
Do[pa=paX[e];pb=pbX[a,b,e];pd=pdX[c,d,e];
If[(f=Sqrt[(pb-pd).(pb-pd)];IntegerQ[f])&&(ar=area[a,b,c,d,e,f]; IntegerQ[ar])&&convexQ[pb,pd,e]&&canonicalQ[{a,b,c,d,e,f}],cnt++
(*;Print[{{a,b,c,d,e,f,ar},Graphics[Line[{pa,pb,pc,pd,pa}]]}]*)],{b,1,a},{e,a-b+1,a+b-1},{c,1,a},{d,Abs[e-c]+1,Min[a,e+c-1]}];
AppendTo[an,cnt],{a,1,85}]
an

cp's OEIS Frontend

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