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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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    Since you only listed 3 terms of the sequence and no recurence relation or formula, there are uncountably many closed forms for it.2012-02-14
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    @N.S.,for example ?2012-02-14
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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.2012-02-14
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    @pejda Check my answer....2012-02-14
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    @KannappanSampath,$F_2(132) \mid S_4 ~ ,F_3(132) \mid S_{10}~,F_5(132) \mid S_{55}$ , where $F_n(132)$ is a Generalized Fermat number with base $132$, and $S_i$ is sequence defined by recurrence relation...2012-02-14
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    @N.S. There are only countably many closed forms for _anything_. (There are uncountably many _sequences_, but most of those have no closed forms).2012-02-14
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    @Pejda that's a completely different problem, since that definition works for many other numbers than $2,3,5$...2012-02-14
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    @HenningMakholm There are uncountably many real numbers, thus there are uncountably many constant sequences, or polynomial closed forms, aren't them?2012-02-14
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    @N.S.: Most of the uncountably many real numbers don't _themselves_ have any closed-form expressions.2012-02-14
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    @N.S.,I am trying to link indexes of $F_n(132)$ , and $S_i$...2012-02-14
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    @HenningMakholm Well it depends what you really mean by a closed form..Is an expression of the form $a_n=an$ a closed form or not for a real number $a$? I say it is, the problem if $a$ is a computable number is different though....2012-02-14
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    [OEIS](http://oeis.org/search?q=%3F%2C4%2C10%2C%3F%2C+55) accepts wild cards and lists a few possible sequences. A nice one is the last one: 2, 4, 10, 40, 55, 162, ..., Integers n such that $8*10^n+21$ is prime.2012-02-14
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    And what is $S_i$? I guess you want to identify $i$....2012-02-14
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    @N.S., $S_i$ is another sequence defined by recurrence relation..I am trying to express $i$ in terms of $n$...2012-02-14
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    Without $a_1,a_4$ it would be $$a_n = \sum\limits_{i=1}^{a_{n-1}}i.$$2012-02-14
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    That is probably exactly what he is looking for Ilya, since he only seems interested in $a_p$...2012-02-14
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    @N.S. please use "@" before the name, otherwise I won't be pinged; moreover, it would avoid the confusion - now it seems that he is looking for me :)2012-02-14
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    @Ilya,But how can I express $4$ using that closed form expression ?2012-02-14
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    @pedja: sorry, I was going to write that my sequence has to be initiated at $4$.2012-02-14
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    @Ilya,@N.S.,related : http://math.stackexchange.com/q/109255/156602012-02-14
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    @pedja: I'm not a number theory person, so I can't help. Hope that my ideas was useful. Regards2012-02-15
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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"...