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$x(0)=1$

$x(1)=41 $

$x(n+2)=3x(n)+\sqrt{8(x(n)^2+x(n+1)^2)}$,

$n\geq0 $ prove that the elements are natural numbers.

The solution: Ax(0)+Bx(1)+Cx(2)=x(3) ; Ax(1)+Bx(2)+Cx(3)=x(4); Ax(2)+Bx(3)+C*x(4)=x(5); You can calculate A,B,C - and these are natural numbers. The biggest problem was that from first shape i thought that it is just second-degree recursion. Thank you your answer, I hope this solution is good.

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    Your first question may have a solution with some clever trick, but in general, recurrence relations of that type are called [quadratic maps](http://mathworld.wolfram.com/QuadraticMap.html) and don't have closed form solutions. Are you certain that it does have a solution? – 2017-01-11
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    One of the rules of this community is to ask a single question per post; you ask two. Please delete any of your questions and ask it again in a separate post. – 2017-01-12
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    Why on earth are you repeating @CodeLabMaster's comment? – 2017-01-14

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