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A236539
Integer areas of integer-sided triangles such that the length of two sides are Fibonacci numbers.
0
6, 12, 30, 60, 126, 396, 480, 840, 1452, 3120, 3696, 5040, 7332, 21840, 47520, 50460, 52440, 66000, 70980, 148512, 178608, 185472, 483360, 1021020, 1025640, 1493280, 2299920, 2772480, 2985234, 3357900, 4442256, 5362980, 6990060, 8233680, 9801000, 14058330
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
Cf. A000045.
Sequence in context: A322374 A351694 A014131 * A122211 A015801 A073245
KEYWORD
nonn
AUTHOR
Michel Lagneau, Jan 28 2014
STATUS
approved