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%I #12 Dec 29 2023 12:59:17
%S 24,24,24,62208,24,30,18,46656,30,24,18,46656,30,30,12,31104
%N Special triangle sides and areas of triangles that transform as Weierstrass elliptic in structure based on the formula A=s*(s-a)*(s-b)*(s-c): s=36.
%C Starting with the formula: A=s*(s-a)*(s-b)*(s-c)=w^2=(ds/dt)^2 a transform set is used to get: A1=4*(s1-a1)*(s1-b1)*(s1-c1)=w1^2=(ds1/dt)^2 w1=2*w/sqrt(s) s1=s/3 a1=a-2*s/3 b1=b-2*s/3 c1=c-2*s/3. The s=36 is found from solving the differential equation in w and w1 with the known condition that s=(a+b+c)/2. The solution set has four integer-sided solutions.
%F {a(n),a(n+1),a(n+2),a(n+3)}={a,b,c,A }[n]
%e a+b+c = 24 + 24 + 24 = 2*s = 72: Area=62208 (equilateral)
%e a+b+c = 24 + 30 + 18 = 2*s = 72: Area=46656 (3,4,5 triangle)
%e a+b+c = 30 + 30 + 12 = 2*s = 72: Area=31104 (the surprise: a 5,5,2 triangle has minimum area)
%t n1[n_] = 18 + n; m1[m_] = 18 + m; l1[n_, m_] = 72 - m1[m] - n1[n]; C0 = Delete[Union[Flatten[Union[Table[Table[If[s*(s - n1[n])*(s - m1[m])*(s - l1[n, m]) > 0 && n1[n]*m1[m]*l1[n, m] > 0, {n1[n], m1[m], l1[n, m], s*(s - n1[n])*(s - m1[m])*(s - l1[n, m])}, {}], {n, 1, 24}], {m, 1, 24}]], 1]], 1] Flatten[C0]
%K nonn,tabf,more,uned
%O 0,1
%A _Roger L. Bagula_, Feb 24 2006
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