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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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IMHO, there is not sufficient data. And, even OEIS seems to know only two sequences containing the three integers you have told us. Please tell us where did you come across this? Some context could help. – user21436 Feb 14 '12 at 17:45
@N.S. There are only countably many closed forms for anything. (There are uncountably many sequences, but most of those have no closed forms). – Henning Makholm Feb 14 '12 at 17:51
@Pejda that's a completely different problem, since that definition works for many other numbers than $2,3,5$... – N. S. Feb 14 '12 at 17:52
@HenningMakholm There are uncountably many real numbers, thus there are uncountably many constant sequences, or polynomial closed forms, aren't them? – N. S. Feb 14 '12 at 17:53
@N.S.: Most of the uncountably many real numbers don't themselves have any closed-form expressions. – Henning Makholm Feb 14 '12 at 17:55
up vote 2 down vote accepted

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"...

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