Find an equation of the sequence $1, 3, 12, 60, 360, 2520, \cdots $. [closed]

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Proof:By rule of formation$$a_1=1=\frac{(1+1)!}{2}$$

$$a_2=3=\frac{(1+2)!}{2}$$

$$a_3=12=\frac{(1+3)!}{2}$$

$$a_4=60=\frac{(1+4)!}{2}$$

$$a_5=360=\frac{(1+5)!}{2}$$

$$a_6=2520=\frac{(1+6)!}{2}$$

$$\vdots$$

$$a_n=\frac{(1+n)!}{2}$$

Is there any other way?

Would appreciate your guidance.

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2 Answers

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When you have a sequence of integers like $1, 3, 12, 60, 360, 2520$ and you want to know what is a likely extension of it... consult theOn-line Encyclopedia of Integer Sequences. Type your sequence in the search box, and see what happens!

In this case, we get 4 sequences that match it. But A00170 is the most likely. (The other three turned out to be essentially this one anyway.)

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"Is there any other way?" Any other way for what? The n-th term is a constant times the factorial of $n+1,$ and the factorial function this has no exact closed-form formula that equals it. You could approximate it by Stirling's approximation, or other methods, but this will only be an approximation; it won't give you the exact value in the sequence.

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