<s>[INST] Find all positive integers $n$ for which $1 + 5a_n.a_{n + 1}$ is a perfect square. The sequence $a_1, a_2, \ldots $ is defined by the initial conditions $$a_1 = 20;  \quad a_2 = 30$$ and the recursion
$$a_{n+2} = 3a_{n+1} - a_n$$  and
for $n \geq 1$. Find all positive integers $n$ for which $1 + 5a_n * a_{n+1}$ is a perfect square.
I could only find the $n$-th term and don't know how to proceed further.pls help 
 [/INST] The only such $n$ is $n=3$, with
$$
1 + 5 a_3 a_4 = 1 + 5 \cdot 70 \cdot 180 = 63001 = 251^2.
$$
Let $b_n = a_n/10 = 2, 3, 7, 18, 47, \ldots$ for $n=1,2,3,4,5,\ldots$ .
These are sums of consecutive odd-order Fibonacci numbers:
$2 = 1+1$ (with the first $1$ being $F_{-1}$),
$3 = 1+2$, $7 = 2+5$, $18 = 5+13$, $47 = 13+34$, etc. by induction.
It soon follows that $b_n b_{n+1} = 5 F^2 + 1$ where $F$ is the
Fibonacci number common to $b_n$ and $b_{n+1}$:
$$
2\cdot 3 = 5 \cdot 1^2 + 1,\phantom{M}
3\cdot 7 = 5 \cdot 2^2 + 1,\phantom{M}
7\cdot 18 = 5 \cdot 5^2 + 1,\phantom{M}
18\cdot 47 = 5 \cdot 13^2 + 1,
$$
etc.
So we're looking to make
$$
1 + 5 a_n a_{n+1} = 1 + 500 b_n b_{n+1} = 2500 F^2 + 501
$$
a square, and it's easy to see that $F = 5$ is the only positive integer
that makes this happen even without the hypothesis that $F$ be
a Fibonacci number.  (For instance, if $2500 F^2 + 501 = y^2$ with $y>0$,
we may factor $501 = y^2 - 2500F^2 = (y-50F) (y+50F)$,
or bound $y$ between $50F$ and $50F+1$ once $F>5$,
or use the technique I described in
this Mathoverflow answer.)  Therefore $n=3$ is the unique answer as claimed.
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