A051516 -id:A051516 - cp's OEIS frontend

A385872 Areas of nondegenerate triangles with perimeter A385737(n) whose side lengths are triangular numbers.

1452, 1176, 2376, 3780, 8316, 10626, 14742, 28500, 12558, 32340, 25200, 94500, 18792, 130680, 89250, 158760, 130680, 155250, 53508, 93636, 122958, 208278, 893970, 1199772, 2183328, 1130976, 2058210, 1414098, 3160080, 4000752, 3898800, 324324, 4900500, 1845120, 7427970

Offset: 1

Felix Huber, Jul 18 2025

nonn

45189144 is the smallest integer area of a right triangle whose sides are triangular numbers. This area corresponds to the triangle [8778, 10296, 13530].
From David A. Corneth, Jul 18 2025: (Start) If sidelengths are u, v, w where 0 < u < v < w < u + v then the area can be written as A = ((u + v + w) * (u + v - w) * (u - v + w) * (-u + v + w)) / 16 = k^2. If A is a square then 16*A is a square (possible extraneous resulting from this can be removed at the end).
We may rewrite 16*A as ((u + v)^2 - w^2) * (w^2 - (v - u)^2) = k^2
Since their product is a square we may write
((u + v)^2 - w^2) * t^2 = (w^2 - (v - u)^2). where t > 1 is a rational. When u, v and t are chosen we can solve for w.
w^2 = (t^2*(u-v)^2 + (u+v)^2) / (t^2 + 1). (End)

			a(1) = 1452 is the area of the triangle [55, 55, 66] with perimeter A385737(1) = 176, where 55 and 66 are triangular numbers.
a(2) = 1176 is the area of the triangle [28, 91, 105] with perimeter A385737(2) = 224, where 28, 91 and 105 are triangular numbers.
From _David A. Corneth_, Jul 18 2025: (Start)
From (u, v) = (28, 91) we get
((u + v)^2 - w^2) * t^2 = (w^2 - (v - u)^2)
(119 - w^2) * t^2 = (w^2 - 63^2). Testing t = 2/3 gives the desired w. (End)

Felix Huber, Table of n, a(n) for n = 1..315
Felix Huber, Maple program
Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A051516, A188158, A385736, A385737.

A385872:=proc(P) # To get all integer areas of triangles with perimeters <= P.
    local p,x,y,z,u,v,w,s,i;
    p:=[];
    for z to floor((sqrt(24*P+9)-3)/6) do
        for x from z to floor((sqrt(4*P-3)-1)/2) do
            for y from max(z,floor((sqrt(1+4*(x^2+x-z^2-z))-1)/2)+1) to min(x,floor((sqrt(1+4*(2*P-x^2-x-z^2-z))-1)/2)) do
            	u:=z*(z+1)/2;
            	v:=y*(y+1)/2;
            	w:=x*(x+1)/2;
            	s:=(u+v+w)/2;
            	if issqr(s*(s-u)*(s-v)*(s-w)) then
               	    p:=[op(p),[u+v+w,sqrt(s*(s-u)*(s-v)*(s-w))]]
               	fi
            od
        od
    od;
    return seq(sort(p)[i,2],i=1..nops(p))
end proc;
A385872(16236);

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

A330917 Largest possible side length, a, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

Original entry on oeis.org

3, 5, 5, 6, 5, 10, 10, 8, 13, 11, 15, 16, 15, 7, 15, 20, 11, 17, 20, 20, 19, 15, 25, 26, 22, 25, 30, 29, 32, 25, 30, 25, 35, 25, 30, 39, 40, 39, 33, 34, 40, 45, 48, 38, 35, 51, 50, 53, 41, 52, 34, 43, 29, 55, 50, 35, 39, 57, 60, 65, 55, 64, 51, 65, 65, 60, 68, 61, 70, 65

Offset: 1

Wesley Ivan Hurt, May 03 2020

nonn

			a(1) = 3; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its shortest side is 3.
a(6) = 10; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], whose smallest side lengths are 4 and 10. The largest of these is 10.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916, A330921.
Cf. A330923, A331199.

A330923 Largest possible side length, b, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

Original entry on oeis.org

4, 5, 5, 8, 12, 13, 13, 15, 15, 13, 17, 17, 25, 24, 25, 29, 25, 25, 25, 29, 20, 26, 30, 35, 26, 40, 39, 40, 41, 40, 51, 33, 48, 38, 50, 45, 58, 41, 60, 51, 65, 65, 61, 60, 56, 68, 65, 75, 50, 72, 61, 61, 60, 74, 80, 84, 68, 65, 87, 89, 90, 82, 87, 80, 89, 102, 100, 74

Offset: 1

Wesley Ivan Hurt, May 03 2020

nonn

			a(1) = 4; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its middle side is 4.
a(6) = 13; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], whose middle side lengths are 13 and 10. The largest of these is 13.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916, A330921.
Cf. A330917, A331199.

A331199 Largest possible side length, c, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

Original entry on oeis.org

5, 6, 8, 10, 13, 15, 17, 17, 20, 20, 21, 24, 26, 25, 29, 30, 30, 26, 29, 35, 37, 37, 39, 41, 40, 41, 45, 48, 48, 51, 53, 52, 53, 51, 58, 60, 61, 50, 65, 65, 68, 70, 74, 74, 75, 75, 78, 80, 73, 82, 75, 68, 85, 87, 89, 89, 87, 87, 95, 97, 97, 97, 101, 102, 104, 106

Offset: 1

Wesley Ivan Hurt, May 03 2020

nonn

			a(1) = 5; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its largest side length is 5.
a(6) = 15; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], whose largest side lengths are 15 and 12. The largest of these is 15.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.
Cf. A330912, A330915, A330916, A330921.
Cf. A330917, A330923.

A070208 Number of integer triangles with perimeter n having integral area but not integral inradius.

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

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051516(n) - A070201(n).

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Heron's Formula.
R. Zumkeller, Integer-sided triangles

A308091 Sum of the areas of the integer-sided triangles with perimeter n and integer area.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 12, 0, 12, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 30, 0, 72, 0, 0, 0, 198, 0, 0, 0, 60, 0, 126, 0, 66, 0, 0, 0, 288, 0, 180, 0, 0, 0, 360, 0, 84, 0, 0, 0, 330, 0, 0, 0, 648, 0, 132, 0, 204, 0, 420, 0, 876, 0, 0, 0, 114

Offset: 1

Wesley Ivan Hurt, May 12 2019

nonn

Wikipedia, Integer Triangle

Cf. A051516.

Table[Sum[Sum[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)] (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}]

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} m * (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)).

A385819 Numbers k such that there are least five primitive Heron triangles having the same area and perimeter k.

Original entry on oeis.org

2842, 3542, 5642, 5750, 6314, 7238, 7546, 9790, 15470, 15778, 17710, 20026, 21658, 21970, 22610, 26962

Offset: 1

Zhining Yang, Jul 09 2025

			3542 is a term because there exists 5 primitive Heron triangles: {{421,1518,1603}, {511,1375,1656}, {583,1288,1671},{759,1096,1687}, {851,1001,1690}} with same perimeter 3542 and same area 318780.
20026 is a term because there exists 6 primitive Heron triangles: {{2108,8493,9425}, {2173,8398,9455}, {2261,8277,9488}, {2418,8075,9533}, {4123,6205,9698}, {4588,5729,9709}} with same perimeter 20026 and same area 8410920.

BBS Math Blog, Primitive Heron triangles with equal perimeter and area (in Chinese).

Cf. A051516, A070138, A188158

sol = Association[];
For[n = 2, n <= 6000, n += 2,
For[z = Ceiling[n/3], z < Floor[n/2], z++,
For[x = 1, x < Floor[n/3], x++, y = n - x - z;
   If[x + y > z > y > x && GCD[x, y, z] == 1, p = (x + y + z)/2;
    A = Sqrt[p (p - x) (p - y) (p - z)];
    If[IntegerQ[A], d = ToString@n <> "->" <> ToString@A; t = {x, y, z};
     If[KeyExistsQ[sol, d], AppendTo[sol[d], t], sol[d] = {t}]]]]]];
Select[sol, Length@# > 4 &]

A330922 Largest possible area of a Heronian triangle with perimeter A051518(n).

Original entry on oeis.org

6, 12, 12, 24, 30, 48, 60, 60, 84, 66, 108, 120, 126, 84, 150, 192, 132, 204, 210, 240, 114, 156, 300, 336, 264, 360, 432, 420, 480, 468, 540, 330, 588, 456, 600, 756, 768, 780, 726, 816, 840, 972, 1080, 456, 924, 1170, 1200, 1260, 984, 1344, 1020, 1290, 522, 1452

Offset: 1

Wesley Ivan Hurt, May 02 2020

nonn

			a(1) = 6; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its area is 3*4/2 = 6.
a(6) = 48; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12], with areas 24 and 48. The largest area of the two triangles is 48.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.

Cf. A330912, A330915, A330916, A330921.

A334677 Number of integer-sided triangles with nonintegral area and perimeter A009005(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 4, 2, 5, 4, 7, 4, 8, 6, 10, 8, 12, 10, 14, 11, 16, 14, 19, 16, 21, 18, 24, 19, 27, 24, 30, 23, 33, 30, 37, 32, 40, 35, 44, 39, 48, 44, 52, 45, 56, 50, 61, 56, 65, 57, 70, 64, 75, 70, 80, 72, 85, 80, 91, 80, 96, 90, 102, 95, 108, 100, 114, 103, 120, 114

Offset: 1

Wesley Ivan Hurt, May 08 2020

nonn

			a(1) = 1; There is one integer-sided triangle with perimeter A009005(1) = 3 whose area is not an integer, [1,1,1] with area sqrt(3)/4.
a(2) = 1; There is one integer-sided triangle with perimeter A009005(2) = 5 whose area is not an integer, [1,2,2] with area sqrt(15)/4.

Wikipedia, Integer Triangle

Cf. A005044, A009005, A051516, A051518.

cp's OEIS Frontend

A385872 Areas of nondegenerate triangles with perimeter A385737(n) whose side lengths are triangular numbers.

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

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Mathematica

A330917 Largest possible side length, a, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

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A330923 Largest possible side length, b, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

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A331199 Largest possible side length, c, of a Heronian triangle with perimeter A051518(n), such that a <= b <= c.

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A070208 Number of integer triangles with perimeter n having integral area but not integral inradius.

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A308091 Sum of the areas of the integer-sided triangles with perimeter n and integer area.

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Mathematica

Formula

A385819 Numbers k such that there are least five primitive Heron triangles having the same area and perimeter k.

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Mathematica

A330922 Largest possible area of a Heronian triangle with perimeter A051518(n).

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A334677 Number of integer-sided triangles with nonintegral area and perimeter A009005(n).

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