# General solution of recurrence relation [closed]

I am supposed to solve for the general solution of $f(n+2)=2(f(n+2))^2 -f(n+2)f(n)-2012$. I tried the method of generating functions but I am stuck with the power $2$ on the RHS. any other methods or idea on how to proceed?

Edit: $f:\mathbb N \to \mathbb N$

## closed as unclear what you're asking by Did, Yagna Patel, yoknapatawpha, user228113, PragabhavaJan 28 '16 at 0:27

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• Any other conditions on $f$? The equation itself gives two values for $f(n+2)$ for each $f(n)$ since $2x^2+ax-2012$ has always two distinct roots. – Mohsen Shahriari Jan 27 '16 at 18:34
• In your case, $a$ is $f(n)+1$. Just subtract $f(n+2)$ from both sides of the equation. – Mohsen Shahriari Jan 27 '16 at 18:42
• With your new condition added, the task is to find out when is the root of the quadratic polynomial a natural number. – Mohsen Shahriari Jan 27 '16 at 19:01
• Until you come with another version, don't vandalize the current one, out of respect for Mohsen Shahriari's work and mine. Otherwise, our answers will look out of context. – Alex M. Jan 27 '16 at 22:48

## 2 Answers

Let $f:\mathbb N\to\mathbb N$ satisfy $$2f(n+2)^2-(f(n)+1)f(n+2)-2012=0$$ for every natural number $n$. Since $f(n+2)$ can't be negative, we have: $$f(n+2)=\frac{f(n)+1+\sqrt{(f(n)+1)^2+16096}}4$$ Now, $f(n+2)$ must be a natural number, so $(f(n)+1)^2+16096$ must be a perfect square. Hence there is a natural number $k$ such that: $$(f(n)+1)^2+16096=k^2$$ $$\therefore\quad k^2-(f(n)+1)^2=16096$$ $$\therefore\quad (k-f(n)-1)(k+f(n)+1)=2^5\cdot503$$ Because $k-f(n)-1$ and $k+f(n)+1$ have the same parity and their product is even, therefore both of them have to be even. Also because $f(n+2)=\frac{f(n)+1+k}4$, therefore $k+f(n)+1$ must be divisible by $4$. Obviously $k+f(n)+1>k-f(n)-1$ so by the above factorization, $(k-f(n)-1,k+f(n)+1)\in\{(8,2012),(4,4024),(2,8048)\}$ which yields $(f(n),f(n+2))\in\{(1001,503),(2009,1006),(4022,2012)\}$. But this is impossible since $(f(n+2),f(n+4))$ must be an element of the same set and hence $f(n+2)\in\{1001,2009,4022\}\cap\{503,1006,2012\}=\varnothing$ which leads to a contradiction. So no such function exists.

As it is formulated now, your problem has no solution.

Note that the relation that you give (that does not specify the initial terms $f(0)$ and $f(1)$ - but this has no consequences over our reasoning) can be rewritten as

$$2012 = f(n+2) \Big( 2 f(n+2) - f(n) - 1 \Big) ,$$

which shows that $f(n+2) \mid 2012, \ \forall n \ge 0$. But $2012 = 4 \cdot 503$, which means that $f(n+2) \in \{ 1, 2, 4, 503, 1006, 2012 \} \ \forall n \ge 0$.

On the other hand, the given relation can be rearranged as a 2nd degree equation in $f(n+2)$ as

$$2 f(n+2) ^2 - \Big( f(n) + 1 \Big) f(n+2) - 2012 = 0 ,$$

which has for discriminant the quantity $\Big( f(n) + 1 \Big)^2 + 16096$. In order for the "solution" $f(n+2)$ to be a natural number, the discriminant must be a perfect square. If $n \ge 2$ then $f(n)$ must be one of the six divisors listed above, but none of them gives a perfect square when plugged in the expression of the discriminant, which shows that your problem has no solution. Are you sure that you have copied it correctly?

• I see no important difference between my answer and yours. – Mohsen Shahriari Jan 28 '16 at 15:23
• @MohsenShahriari: I was typing mine while yours was deleted. When I published it, I discovered that you had already undeleted yours too, without me knowing this (yours was edited 17 hours ago, mine was answered 17 hours ago - at the time of writing this comment - so we were working almost simultaneously). Given that I had spent some time thinking about it, I decided not delete it. I also feel it is less convoluted than yours. In any case, it seems like the OP has deleted his account, so we won't get any feedback on this anymore. – Alex M. Jan 28 '16 at 15:47