A316853 -id:A316853 - cp's OEIS frontend

A371972 a(n) is the number of distinct areas of triangles with integer sides whose largest side is n.

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 120, 131, 144, 156, 168, 182, 196, 210, 225, 239, 256, 270, 288, 306, 321, 342, 361, 380, 399, 420, 441, 460, 484, 505, 527, 552, 576, 599, 623, 649, 673, 702, 729, 752, 781, 808, 840, 870, 900

Offset: 1

Hugo Pfoertner, Apr 16 2024

nonn

Cf. A316841, A371973, A371345.

See the formula section for the relationships with A002620, A173196, A316843, A316853.

A2(a,b,c) = {my(s=(a+b+c)/2);s*(s-a)*(s-b)*(s-c)};
a371972(n) = {my (A=List()); for(s2=1,n, for(s3=1,s2, if(s2+s3>n, listput(A, A2(n,s2,s3))))); #Set(A)};

a(n) <= A002620(n+1), with equality for n <= 20.

a(n) = |{A316853(k) : A316843(k) = n}| = |{A316853(k) : A173196(n) < k <= A173196(n+1)}|. - Peter Munn, Jul 30 2025

A383413 Area A of triangles such that the sides are distinct integers and A is an integer.

Original entry on oeis.org

6, 24, 30, 36, 42, 54, 60, 66, 72, 84, 90, 96, 114, 120, 126, 132, 144, 150, 156, 168, 180, 198, 204, 210, 216, 234, 240, 252, 264, 270, 288, 294, 300, 306, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 456, 462, 468, 480, 486, 504, 510, 522, 528, 540, 546, 576, 594

Offset: 1

Karl-Heinz Hofmann, Apr 26 2025

nonn

All terms are multiples of 6.

Subsequence of A188158.

			72 is in the sequence because the triangle with sides {a=5, b=29, c=30} has an area of exactly 72 and all sides are distinct.
12 is not in the sequence because this area is only possible with the isosceles triangles {a=5, b=5, c=6} and {a=5, b=5, c=8} with a and b not distinct.

Wikipedia, Heronian triangle

Cf. A188158, A316853.

nn = 450; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[a!=b!=c&&0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a-1}, {c, b-1}]; Union[lst]; lst (* James C. McMahon, May 10 2025 *)

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A371972 a(n) is the number of distinct areas of triangles with integer sides whose largest side is n.

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A383413 Area A of triangles such that the sides are distinct integers and A is an integer.

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