What is the pattern here?

Question

a1=q
a2=q + 2q^2
a3=q + 6 q^2 + 2q^3
a4=q + 14 q^2 + 12 q^3
a5=q + 30 q^2 + 50 q^3
a6=???

Answer 1

This is out there, but I think it's

a6 = q + 62 q^2 + 180 q^3

although I'd have thought there would already be q^4 terms. The integers are related to "Stirling numbers of the second kind," which can be viewed in a triangle like Pascal's.

First diagonal: 1 1 1 1 1 1...
Second: 1 3 7 15 31 63... (subtract one for q^2 coef)
Third: 1 6 25 90 301... (double for q^3)

just in case, fourth: 1 10 65 350 1701...

For more info and lots of links, see the Online Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/A008277

Answer 2

The q^2 term is increasing by adding 4 then adding 8 then adding 16 so the unknown would be derived by adding 32 to 30 for 62q^2.

The q^3 term is stumping me.


****How did you get 152 bedbye? What's the pattern?

Answer 3

Partial help

a6 = q + 62q^2 + ??q^3

You can definitely see the pattern for the coefficient of q^2. The difference follows a geometric progression:
2,4,8,16

I'll try to work on the q^3

Answer 4

a6 = q + 62q^2 + 152q^3

Answer 5

??