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Please help me solve this weird recurrence relation. This is not really standard quadratic, so I'm totally confused. I tried with logarithm (but 8 is excess), tried writing this recurrence in one degree up or down... I cannot sleep for days because of this :-( This is what troubles me:$$a_{n+2}a_{n}=a_{n+1}^2+8$$for$$a_1=4$$and$$a_2=6$$ Please just give idea or complete solution (step by step) - I'm desperate!(I don't know how write square, so please edit. Thanks.)

Joelafrite
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Lara Ljumović
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    What is $a_n a_{n+2} - a_{n+1}^2$ if $a_n = B^n + C^n?$ – Will Jagy May 17 '15 at 00:04
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    That way is almost good enough; what, precisely, is $a_n a_{n+2} - a_{n+1}^2$ if $a_n = \beta B^n + \gamma C^n? $ Then, how do you get $a_n a_{n+2} - a_{n+1}^2$ to become constant? – Will Jagy May 17 '15 at 00:25
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    Sigh; got to sleep, evidently. This type of relationship is discussed in Conway and Guy, The Book of Numbers. Probably chapter 3, and called a Number Wall. – Will Jagy May 17 '15 at 00:38
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    @willjagy I am still up and listening my friend. – Chinny84 May 17 '15 at 00:45

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