Generating function for $15, 66,189,420,795,\dots$

Question

I am to derive the formula for $15,66,189,420, 795,\dots$

I have the common pattern as $3m(2n+1)$ where $n$ and $m$ are integers. For example,

$66=3\cdot2\cdot(2\cdot5+1)$

Will this end up in me deriving the correct formula or is there another pattern? Also could someone help me with the next step.

That's what the question asks,not sure what is meant by where the numbers are from — Kadoe, Nov 11 '16 at 23:00

score 2 · Answer 1 · answered Nov 11 '16 at 23:08

2

You seem to be on the right track. The numbers are

$$3\cdot 1\cdot 5$$

$$3\cdot 2\cdot 11$$

$$3\cdot 3\cdot 21$$

$$3\cdot 4\cdot 35$$

$$3\cdot 5\cdot 53$$

The differences in the last column are $6,10,14,18$ , forming an arithmetic progression. So, we have $$a_n=3n\cdot (2n^2+3)=6n^3+9n$$

answered Nov 11 '16 at 23:08

Peter

1

Well played.... – Evariste Nov 11 '16 at 23:10
Thank you, Evariste! The main work was done by the OP, though. – Peter Nov 11 '16 at 23:11
Thank you so much, after this i just used scaling and differentiate and other methods to derive a function correct? – Kadoe Nov 11 '16 at 23:54

1 Answers1