Question

What is the next number 6,9,13,20,31

Answer 1

`a_n=\{6,9,13,20,31,\ldots\}`

Now, look at this

`a_1=6\\a_2=9=a_1+3\\a_3=13=a_2+4\\a_4=20=a_3+7\\a_5=31=a_4+11`

and notice, that

`a_4=a_3+a_3+a_2-a_2-a_1\\a_5=a_4+a_4+a_3-a_3-a_2`

which can be simplified to

`a_4=2a_3-a_1\\a_5=2a_4-a_2`

Therefore, we can deduce that

`a_n=2a_(n-1)-a_(n-3)`

So

`a_6=2a_5-a_3=2\cdot31-13=62-13=49`

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After I posted the above answer, I've noticed another relationship between the numbers in the sequence

`a_1=6,a_2=9\\a_3=13=9+6-2=a_2+a_1-2\\a_4=20=13+9-2=a_3+a_2-2\\a_5=31=20+13-2=a_4+a_3-2`

So

`a_n=a_(n-1)+a_(n-2)-2`

Therefore

`a_6=a_5+a_4-2=31+20-2=49`

Which, incidentally (or maybe not?) the sequence appears to be a subsequence of another sequence that is known as A000211 in the OEIS.