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I am trying to find closed form expression of following sequence :

$ a_1,4,10,a_4,55,...$

In other words what would be n'th term of this sequence ?

I have tried few recurrence relations but couldn't find adequate one .

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    @Ilya,Thanks anyway......2012-02-15

1 Answers 1

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If those are the only terms you know, then here is what you can do

Step 1 Let $f(x)=ax^2+bx+c$. Solve the system $f(2)=4, f(3)=10, f(5)=55$. Since the determinant of the system is vandermonde, this system has unique solution [ You can also check lagrange Interpolation Polynomial instead, in this case it is exactly the same thing].

Step 2 Pick $g$ any function defined on the positive integers. Then

$a_n=f(n)+g(n)(n-2)(n-3)(n-5) \,.$

is a "closed formula"...