A024153 -id:A024153 - cp's OEIS frontend

A188158 Area A of the triangles such that A and the sides are integers.

6, 12, 24, 30, 36, 42, 48, 54, 60, 66, 72, 84, 90, 96, 108, 114, 120, 126, 132, 144, 150, 156, 168, 180, 192, 198, 204, 210, 216, 234, 240, 252, 264, 270, 288, 294, 300, 306, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 432, 456, 462, 468, 480, 486, 504, 510, 522, 528

Offset: 1

Michel Lagneau, Mar 22 2011

nonn

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2. A given area often corresponds to more than one triangle; for example, a(9) = 60 for the triangles (a,b,c) = (6,25,29), (8,17,15), (13,13,10) and (13,13,24).

If only primitive integer triangles (that is, the lengths of the sides are coprime) are considered, then the possible areas are 6 times the terms in A083875. - T. D. Noe, Mar 23 2011

			a(3) = 24 because the area of the triangle whose sides are 4, 15, 13 is given by sqrt(p(p-4)(p-15)(p-13)) = 24, where p = (4 + 15 + 13)/2 = 16.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangle
Wikipedia, Heronian triangle

Cf. A007237, A009112, A024153, A024365, A051516, A051584, A051585, A055592, A055593, A055594, A055595.

# storage of areas in T(i)
T:=array(1..4000):nn:=100:k:=1:for a from 1
  to nn do: for b from 1 to nn do: for c from 1 to nn do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c):   if x>0 then x1:=abs(x):s:=sqrt(x1) :else fi:if s=floor(s) then T[k]:=s:k:=k+1:else
  fi:od:od:od:
# sort of T(i)
for jj from 1 to k-1 do: ii:=jj:for k1 from  ii+1 to k-1 do:if T[ii]>T[k1] then ii:=k1:else fi:od: m:=T[jj]:T[jj]:=T[ii]:T[ii]:=m:od:liste:=convert(T,set):print(liste):
# second program:
isA188158 := proc(A::integer)
    local Asqr, s,a,b,c ;
    Asqr := A^2 ;
    for s in numtheory[divisors](Asqr) do
        if s^2> A then
        for a from 1 to s-1 do
            if modp(Asqr,s-a) = 0 then
                for b from a to s-1 do
                    c := 2*s-a-b ;
                    if s*(s-a)*(s-b)*(s-c) = Asqr then
                        return true ;
                    end if;
                end do:
            end if;
        end do:
        end if;
    end do:
    false ;
end proc:
for n from 3 to 600 do
    if isA188158(n) then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, May 02 2018

nn = 528; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst] (* T. D. Noe, Mar 23 2011 *)

A051516 Number of triangles with perimeter n having integer sides and area.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 4, 0, 1, 0, 0, 0, 3, 0, 0, 0, 5, 0, 1, 0, 1, 0, 2, 0, 5, 0, 0, 0, 1, 0, 1, 0, 4, 0, 0, 0, 8, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 5, 0, 6, 0, 5, 0, 0, 0, 2, 0, 0, 0, 12, 0, 1, 0

Offset: 1

David W. Wilson

nonn

No such triangles with odd perimeter exist.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Seiichi Manyama)
Eric Weisstein's World of Mathematics, Heronian Triangle.

Cf. A024153, A070139.

Table[Sum[Sum[(1 - Ceiling[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]] + Floor[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]])*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 11 2019 *)

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - ceiling(m) + floor(m)) * sign(floor((i+k)/(n-i-k+1))), where m = sqrt((n/2)*(n/2-i)*(n/2-k)*(i+k-n/2)). - Wesley Ivan Hurt, May 11 2019

A070139 Number of isosceles integer triangles with perimeter n having integral area.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051516(n) - A024153(n).

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Heronian Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A051516, A024153, A059169, A070145.

A070203 Number of scalene integer triangles with perimeter n having integral inradius.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 8, 0, 0, 0, 1, 0, 3

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A070201(n) - A070204(n).

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Scalene Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A024153, A005044.

def A(n)
  cnt = 0
  (1..n / 3).each{|a|
    (a + 1..(n - a) / 2).each{|b|
      c = n - a - b
      if a + b > c && b < c
        s = n / 2r
        t = (s - a) * (s - b) * (s - c) / s
        if t.denominator == 1
          t = t.to_i
          cnt += 1 if Math.sqrt(t).to_i ** 2 == t
        end
      end
    }
  }
  cnt
end
def A070203(n)
  (1..n).map{|i| A(i)}
end
p A070203(100) # Seiichi Manyama, Oct 07 2017

A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 13, 19, 14, 21, 19, 23, 20, 27, 23, 30, 27, 32, 29, 35, 32, 39, 34, 44, 39, 48, 43, 52, 47, 55, 51, 60, 53, 63, 59, 69, 58, 74, 67, 78, 73, 84, 75, 90, 81, 92, 88, 101, 91, 108, 93, 112, 106

Offset: 3

Hugo Pfoertner, Apr 16 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 3..10000

Cf. A005044, A007237, A024153, A317182, A371070.

See the formula section for the relationships with A026810, A070083, A135622 (which has many crossrefs related to areas of triangles).

A2(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
a371973(n) = {my (A=List()); forpart (v=n, listput(A, A2(v[1],v[2],v[3])), [1,(n-1)\2], [3,3]); #Set(A)};

def A371973(n): return len(set((2*(b+c)-n)*(n-2*b)*(n-2*c) for c in range((n+2)//3, (n+1)//2) for b in range((n-c+1)//2, c+1))) # David Radcliffe, Aug 01 2025

a(n) = |{A135622(k) : A070083(k) = n}| = |{A135622(k) : A026810(n) < k <= A026810(n+1)}|. - Peter Munn, Jul 29 2025

b-file corrected by David Radcliffe, Aug 01 2025

A301384 Number of integer-sided triangles of area A188158(n).

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 2, 1, 4, 1, 1, 4, 2, 2, 2, 1, 4, 3, 1, 2, 1, 2, 4, 2, 2, 1, 1, 6, 3, 1, 5, 3, 3, 2, 2, 1, 4, 1, 2, 4, 8, 5, 1, 2, 1, 3, 1, 15, 2, 4, 2, 1, 5, 1, 6, 2, 1, 3, 4, 3, 1, 2, 2, 1, 2, 4, 5, 1, 5, 4, 1, 2, 3, 5, 1, 1, 1, 6, 2, 4, 2, 1, 2, 1, 17, 4, 1

Offset: 1

Michel Lagneau, Mar 20 2018

nonn

Nonzero terms of A051584.

			a(9) = 4 because A188158(9) = 60 corresponding to 4 triangles (a, b, c) = (6, 25, 29), (8, 15, 17), (10, 13, 13) and (13, 13, 24) of area 60.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangle

Cf. A188158, A024153 (distinct side lengths), A051516, A051584, A051585.

A301384 := proc(A::integer)
    local Asqr, s,a,b,c,sol ;
    sol := 0 ;
    Asqr := A^2 ;
    for s in numtheory[divisors](Asqr) do
        if s^2> A then
        for a from 1 to s-1 do
            if modp(Asqr,s-a) = 0 then
                for b from a to s-1 do
                    c := 2*s-a-b ;
                    if s*(s-a)*(s-b)*(s-c) = Asqr and c >= b then
                        sol := sol+1 ;
                    end if;
                end do:
            end if;
        end do:
        end if;
    end do:
    sol ;
end proc:
for n from 3 to 600 do
    a301384 := A301384(n) ;
    if a301384 > 0 then
        printf("%d,\n",a301384) ;
    end if;
end do: # R. J. Mathar, May 02 2018

nn=1000;lst={};lst2={};Do[s=(a+b+c)/2;If[IntegerQ[s],area2=s(s-a)(s-b)(s-c);If[0

cp's OEIS Frontend

A188158 Area A of the triangles such that A and the sides are integers.

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A051516 Number of triangles with perimeter n having integer sides and area.

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A070139 Number of isosceles integer triangles with perimeter n having integral area.

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A070203 Number of scalene integer triangles with perimeter n having integral inradius.

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A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

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A301384 Number of integer-sided triangles of area A188158(n).

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