OFFSET
1,1
COMMENTS
Conjecture: there exists only one triangle where three sides are of Fibonacci number length. This triangle is (5, 5, 8), and the area is A = 12.
The following table gives the first values (A, a, b, c)
------------------------
| A | a | b | c |
------------------------
| 6 | 3 | 4 | 5 |
| 12 | 5 | 5 | 8 |
| 30 | 5 | 12 | 13 |
| 60 | 10 | 13 | 13 |
| 126 | 13 | 20 | 21 |
| 396 | 34 | 55 | 87 |
| 480 | 34 | 34 | 60 |
| 840 | 21 | 89 | 100 |
| 1452 | 55 | 55 | 66 |
| 3120 | 78 | 89 | 89 |
........................
MAPLE
with(combinat, fibonacci):nn:=30000:n1:=40:lst1:={}:for i from 1 to n1 do:a:=fibonacci(i): for j from i to n1 do: b:=fibonacci(j):for c from 1 to nn do:p:=(a+b+c)/2:x:=p*(p-a)*(p-b)*(p-c):if x>0 then q:=sqrt(x):if q=floor(q) then lst1:=lst1 union {q}:printf ( "%d %d %d %d \n", q, a, b, c):else fi:fi:od:od:od:print(lst1):
MATHEMATICA
n1 = 30000; n2 = 40; lst = {}; Do[s = (Fibonacci[a] + Fibonacci[b] + c)/2; If[IntegerQ[s], area2 = s (s - Fibonacci[a]) (s - Fibonacci[b]) (s - c); If[0 < area2 && IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, n2}, {b, a}, {c, 1, n1}]; Union[lst]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Jan 28 2014
STATUS
approved