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A115028 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. 0
24, 24, 24, 62208, 24, 30, 18, 46656, 30, 24, 18, 46656, 30, 30, 12, 31104 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..15.

FORMULA

{a(n),a(n+1),a(+2),a(n+3)}={a,b,c,A }[n]

EXAMPLE

a+b+c=24+24+ 24=2*s=72: Area=62208 ( equilateral)

a+b+c=24+ 30+ 18=2*s=72: Area=46656 ( 3,4,5 triangle)

a+b+c=30+ 30+ 12=2*s=72: Area=31104 ( the surprise: a 5,5,2 triangle has minimum area)

MATHEMATICA

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]

CROSSREFS

Sequence in context: A010863 A217140 A235249 * A099543 A188311 A185497

Adjacent sequences:  A115025 A115026 A115027 * A115029 A115030 A115031

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Feb 24 2006

STATUS

approved

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Last modified May 18 13:52 EDT 2021. Contains 343995 sequences. (Running on oeis4.)