General term of the sequence $2, 7, 16, 30, 50, 77, 112, 156$

Question

What should be the general term of the sequence $2, 7, 16, 30, 50, 77, 112, 156$?

Answer 1

Often looking at successive differences is enough (it works for any polynomial sequence -- at some point the differences will be constant, no matter what polynomial generates the sequence).

$2,7,16,30,50,77,112,156$

$7-2=(5), 16-7=(9), 30-16=(14), 50-30=(20),$
$77-50=(27),112-77=(35), 156-112=(44)$

$9-5=[4], 14-9=[5], 20-14=[6],$
$27-20=[7], 35-27=[8], 44-35=[9]$

$5-4=|1|, 6-5=|1|, 7-6=|1|, 8-7=|1|, 9-8=|1|$

Answer 2

Given such a question, always try OEIS: http://oeis.org/A005581

Here, you may write

$$u_n=\frac 16 (n+1)(n+2)(n+6)={n+3 \choose 3}+{n+2\choose 2}$$

Starting at $n=0$.

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Alternately, to discover the formula, use the binomial transform of your sequence $2,7,16,30,50,77,112,156$, yielding

$$2,5,4,1,0,0,0,0$$

Then your sequence is interpolated by

$$2{n\choose 0}+5{n\choose 1}+4{n\choose 2}+{n\choose 3}$$

Also starting at $n=0$.

And you get a polynomial by expanding the binomials

$$2+5n+4\frac{n(n-1)}{2}+\frac{n(n-1)(n-2)}{6}$$

See the link for details about this method.