A070083 -id:A070083 - cp's OEIS frontend

A070200 Inradii of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.

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

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

Triangles [A070080(A070209(n)), A070081(A070209(n)), A070082(A070209(n))] have integer inradii = a(A070209(k))= A070210(k).

			[A070080(25), A070081(25), A070082(25)] = [3,5,6] and s = A070083(25)/2 = (3+5+6)/2 = 7: a(25) = sqrt((s-3)*(s-5)*(s-6)/7) = sqrt((7-3)*(7-5)*(7-6)/7) = sqrt(4*2*1/7) = sqrt(8/7) = 1.069, rounded = 1.

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

Cf. A070086.

a(n) = sqrt((s-u)*(s-v)*(s-w)/s), where u=A070080(n), v=A070081(n), w=A070082(n) and s=A070083(n)/2=(u+v+w)/2.

A070145 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with integer area.

Original entry on oeis.org

39, 52, 269, 372, 375, 862, 957, 972, 1204, 1955, 1971, 1988, 2790, 2796, 3818, 5374, 6522, 6880, 6881, 6921, 7234, 7310, 7341, 7360, 9198, 9207, 10272, 14506, 15101, 15177, 15237, 21289, 21493, 21540, 21552, 21589

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(1)=39: [A070080(39), A070081(39), A070082(39)] = [5=5<6], area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

Eric Weisstein's World of Mathematics, Heronian Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070142, A070144, A070139.

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

A385737 Perimeters of nondegenerate triangles with integer areas, whose side lengths are triangular numbers.

Original entry on oeis.org

176, 224, 264, 336, 504, 644, 756, 950, 1196, 1232, 1280, 1500, 1566, 1650, 1700, 2100, 2112, 2250, 2366, 2754, 3036, 3306, 5676, 5796, 7296, 8064, 8316, 8526, 9576, 10206, 10260, 12474, 13200, 15872, 16236, 16896, 17094, 17150, 20172, 21714, 21726, 22382, 22644

Offset: 1

Felix Huber, Jul 16 2025

nonn

224 and 1280 are the only perimeters <= 10^6 of nondegenerate triangles whose side lengths (28, 91, 105 or 325, 325, 630, respectively) and areas (1176 or 25200, respectively) are triangular numbers.

			176 is a term because it is the perimeter of the triangle [55, 55, 66], where 55 and 66 are triangular numbers, which has an integer area of sqrt(88*(88 - 55)*(88 - 55)*(88 - 66)) = 1452.
224 is a term because it is the perimeter of the triangle [28, 91, 105], where 28, 91 and 105 are triangular numbers, which has an integer area of sqrt(112*(112 - 28)*(112 - 91)*(112 - 105)) = 1176 (which is also a triangular number).

Felix Huber, Table of n, a(n) for n = 1..315
Felix Huber, Maple program to compute the triangles (incl. areas) with perimeter a(n)
Eric Weisstein's World of Mathematics, Triangular Number

Subsequence of A380875.

Cf. A000217, A051516, A070083, A385736, A385872.

A385737:=proc(P) # To get all perimeters <= P.
    local p,x,y,z,u,v,w,s;
    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]
               	fi
            od
        od
    od;
    return op(sort(p))
end proc;
A385737(22644);

A070144 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle with integer area.

Original entry on oeis.org

17, 116, 212, 252, 368, 370, 493, 561, 587, 659, 839, 850, 1156, 1186, 1196, 1297, 1582, 1599, 1629, 1912, 1920, 2115, 2352, 2555, 2574, 2713, 2774, 2778, 3251, 3473, 3728, 3746, 3751, 4286, 4298, 4307, 4313, 4319

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(2)=116: [A070080(116), A070081(116), A070082(116)] = [6<8<10], area^2 = s*(s-6)*(s-8)*(s-10) with s=A070083(116)/2=(6+8+10)/2=12, area^2=12*6*4*2=64*9 is an integer square, therefore A070086(116)=area=8*3=24.

Eric Weisstein's World of Mathematics, Heronian Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070142, A070145.

A070146 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle with integer area.

Original entry on oeis.org

39, 269, 375, 587, 862, 972, 1196, 1955, 1988, 2352, 2555, 2796, 3818, 4319, 4406, 5378, 6522, 6808, 6880, 6890, 6921, 7234, 7360, 8193, 9159, 9207, 10272, 14545, 15004, 15061, 15101, 15216, 15237, 15943, 16502

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(1)=39: [A070080(39), A070081(39), A070082(39)] = [5,5,6]: A070085(39)=5^2+5^2-6^2=14>0 and area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(39)/2=(5+5+6)/2=8, area^2=8*3*3*2=16*9 is an integer square, therefore A070086(39)=area=4*3=12.

Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Acute Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A070142, A070136, A070147, A070140.

A070147 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an obtuse integer triangle with integer area.

Original entry on oeis.org

52, 252, 368, 372, 561, 659, 839, 957, 1156, 1186, 1204, 1582, 1912, 1920, 1971, 2115, 2713, 2774, 2790, 3251, 3473, 3728, 3746, 4286, 4307, 4313, 4330, 5008, 5272, 5374, 6369, 6389, 6432, 6776, 6881, 7223, 7310, 7341

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(1)=52: [A070080(52), A070081(52), A070082(52)] = [5,5,8]: A070085(52)=5^2+5^2-8^2=-14<0 and area^2 = s*(s-5)*(s-5)*(s-6) with s=A070083(52)/2=(5+5+8)/2=9, area^2=9*4*4*1=16*9 is an integer square, therefore A070086(52)=area=4*3=12.

Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Obtuse Triangle.
Reinhard Zumkeller, Integer-sided triangles

Cf. A070142, A070136, A070146, A070141.

A070210 Inradii of integer triangles [A070080(A070209(n)), A070081(A070209(n)), A070082(A070209(n))].

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 2, 4, 3, 4, 4, 3, 2, 4, 5, 3, 6, 4, 6, 6, 6, 4, 6, 3, 4, 3, 6, 4, 5, 4, 3, 6, 5, 7, 8, 6, 4, 6, 8, 7, 8, 9, 3, 9, 5, 6, 9, 8, 10, 6, 6, 6, 9, 8, 4, 8, 9, 7, 10, 6, 10, 12, 6, 12, 12, 5, 3, 7, 8, 10, 4, 9, 10, 11, 6, 12, 3, 6, 9, 12, 12, 7, 8

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A070200(A070209(n)).

			A070209(3)=212: [A070080(212), A070081(212), A070082(212)] = [5,12,13], let s = A070083(212)/2 = (5+12+13)/2 = 15 then inradius = sqrt((s-5)*(s-5)*(s-6)/s) = sqrt(10*3*2/15) = sqrt(4) = 2; a(3) = A070200(212) = 2.

Mohammad K. Azarian, Solution of problem 125: Circumradius and Inradius, Math Horizons, Vol. 16, No. 2 (Nov. 2008), p. 32.
Eric Weisstein's World of Mathematics, Incircle.
R. Zumkeller, Integer-sided triangles

Cf. A070149, A070201.

A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

Original entry on oeis.org

49, 50, 54, 64, 75, 78, 80, 88, 90, 91, 98, 100, 104, 108, 112, 117, 120, 121, 125, 126, 128, 133, 136, 140, 144, 147, 150, 156, 160, 162, 165, 168, 169, 170, 175, 176, 180, 182, 184, 188, 192, 195, 196, 198, 200, 203, 208, 210, 216, 220, 224, 225, 231, 234, 238, 240

Offset: 1

Klaus Nagel and Hugo Pfoertner, Apr 14 2024

nonn

			a(1) = 49 is the perimeter of the first decomposable triangle with sides of the outer triangle [8, 19, 22], and sides meeting at the 4th "inner" vertex: 17, 6, 4. The 3 inner triangles have sides [8, 4, 6], [19, 17, 4], and [22, 6, 17].

These triangles can be viewed as degenerate tetrahedrons, in which all triangular inequalities for the faces are satisfied, and the Cayley-Menger determinant, which determines whether the 4th vertex yields a valid tetrahedron, takes the value 0.

Klaus Nagel, Illustration of a(1) = 49.
Klaus Nagel, Illustration of a(2) = 50.
Klaus Nagel, Illustration of a(3) = 54.
Klaus Nagel, Illustration of a(4) = 64; another triangle is [20, 21, 23].
Hugo Pfoertner, List of triangles up to perimeter 400; one example of each.
Wikipedia, Cayley-Menger determinant

Cf. A070083, A316841, A342720, A371345, A371973.

H(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
CM(w1,w2,w3,v1,v2,v3) = matdet([0,1,1,1,1; 1,0,w3^2,w2^2,v1^2; 1,w3^2,0,w1^2,v2^2; 1,w2^2,w1^2,0,v3^2; 1,v1^2,v2^2,v3^2,0]);
is_a371969(peri) = {forpart (w=peri, my (A=H(w[1],w[2],w[3]), epsA=1e-12); for (v1=1, w[3]-2, for (v2=w[3]-v1+1, w[3]-2, my (A3=H(w[3],v2,v1)); if (A3>=A, next); for (v3=1, w[3]-2, if (v3+v2<=w[1] || v3+v1<=w[2], next); my (A1=H(w[1],v2,v3)); if (A1>=A, next); my (A2=H(w[2],v1,v3)); if (A2>=A, next); my (C=CM(w[1],w[2],w[3],v1,v2,v3)); if (C==0 && abs(sqrt(A)-sqrt(A1)-sqrt(A2)-sqrt(A3)) < epsA,
\\ print (peri," ",Vec(w)," ",[v1,v2,v3]);
return(1))))), [1,(peri-1)\2], [3,3]); 0};
for (n=3, 100, if (is_a371969(n), print1(n,", ")))

A070150 Triangular areas of integer Heronian triangles.

Original entry on oeis.org

6, 36, 66, 120, 36, 120, 120, 210, 210, 120, 300, 210, 210, 300, 378, 630, 528, 780, 528, 210, 630, 630, 300, 1176, 780, 2016, 990, 1176, 2016, 2016, 1596, 780, 1770, 528, 300, 2850, 630, 2016, 780, 990, 3240, 2016, 630

Offset: 1

Reinhard Zumkeller, May 05 2002

a(n) = A070086(A070148(n)).

			A070148(2)=368: [A070080(368), A070081(368), A070082(368)] = [9,10,17], area^2 = s*(s-9)*(s-10)*(s-17) with s=A070083(368)/2=(9+10+17)/2=18, area^2=18*9*8*1=16*81 is an integer square, therefore area=4*9=36=A000217(8).

Eric Weisstein's World of Mathematics, Heronian Triangle.
R. Zumkeller, Integer-sided triangles

maxPerim = 300; maxSide = Floor[(maxPerim - 1)/2]; order[{a_, b_, c_}] := (a + b + c)*maxPerim^3 + a*maxPerim^2 + b*maxPerim + c; triangles = Reap[ Do[ If[ a + b + c <= maxPerim && c - b < a < c + b && b - a < c < b + a && c - a < b < c + a, Sow[{a, b, c}]], {a, 1, maxSide}, {b, a, maxSide}, {c, b, maxSide}]][[2, 1]]; stri = Sort[ triangles, order[#1] < order[#2] &]; area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p*(p - a)*(p - b)*(p - c)]]; triangularQ[n_] := IntegerQ[Sqrt[8*n + 1]]; area /@ Select[stri, IntegerQ[area[#]] && triangularQ[area[#]] &] (* Jean-François Alcover, Feb 22 2013 *)

cp's OEIS Frontend

A070200 Inradii of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.

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A070145 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an isosceles integer triangle with integer area.

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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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A385737 Perimeters of nondegenerate triangles with integer areas, whose side lengths are triangular numbers.

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Maple

A070144 Numbers n such that [A070080(n), A070081(n), A070082(n)] is a scalene integer triangle with integer area.

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A070146 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an acute integer triangle with integer area.

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A070147 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an obtuse integer triangle with integer area.

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A070210 Inradii of integer triangles [A070080(A070209(n)), A070081(A070209(n)), A070082(A070209(n))].

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A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

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PARI

A070150 Triangular areas of integer Heronian triangles.

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