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.

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

Author

Klaus Nagel and Hugo Pfoertner, Apr 14 2024

Keywords

• nonn

Examples

			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].
		

References

• 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.

Links

• 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

Crossrefs

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

Programs

PARI

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,", ")))