cp's OEIS Frontend

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.

%I A371969 #16 Apr 20 2024 05:55:17
%S A371969 49,50,54,64,75,78,80,88,90,91,98,100,104,108,112,117,120,121,125,126,
%T A371969 128,133,136,140,144,147,150,156,160,162,165,168,169,170,175,176,180,
%U A371969 182,184,188,192,195,196,198,200,203,208,210,216,220,224,225,231,234,238,240
%N 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.
%D A371969 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.
%H A371969 Klaus Nagel, <a href="/A371969/a371969.png">Illustration of a(1) = 49</a>.
%H A371969 Klaus Nagel, <a href="/A371969/a371969_1.png">Illustration of a(2) = 50</a>.
%H A371969 Klaus Nagel, <a href="/A371969/a371969_2.png">Illustration of a(3) = 54</a>.
%H A371969 Klaus Nagel, <a href="/A371969/a371969_3.png">Illustration of a(4) = 64</a>; another triangle is [20, 21, 23].
%H A371969 Hugo Pfoertner, <a href="/A371969/a371969.txt">List of triangles up to perimeter 400</a>; one example of each.
%H A371969 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cayley%E2%80%93Menger_determinant#3-Simplex">Cayley-Menger determinant</a>
%e A371969 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].
%o A371969 (PARI) H(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
%o A371969 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]);
%o A371969 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,
%o A371969 \\ print (peri," ",Vec(w)," ",[v1,v2,v3]);
%o A371969 return(1))))), [1,(peri-1)\2], [3,3]); 0};
%o A371969 for (n=3, 100, if (is_a371969(n), print1(n,", ")))
%Y A371969 Cf. A070083, A316841, A342720, A371345, A371973.
%K A371969 nonn
%O A371969 1,1
%A A371969 _Klaus Nagel_ and _Hugo Pfoertner_, Apr 14 2024