# $a_n = 2n^2, n \geq 1$ (modified)

The puzzle asks to determine the next term in the sequence 2, 4, 8, 0, 0, 16...

a. 32
b. 18
c. 20
d. 24
e. 0

The obvious answer for me would be 32, with these zeros serving as the "neutral element" of the sequence. But I feel uncomfortable because it's almost like discarding them to produce a sequence like 2, 4, 8, 0, 0, 16, 32, 64, 0, 0, 128, 256, ...

Is there some closed formula or logic to determine the next term? Maybe an = 0, n mod 5 = 0?

• What does the title refer to? Does the puzzle ask to modify 2n^2 to get 2, 4, 8, 0, 0, 16... that continues to one of a.-e. answers? – Vepir Jan 1 at 18:04
• I feel like the question needs more information/context to be answerable. I agree the pattern of numbers/gaps could be 3-2-3-2..., but I was more interested in the fact the numbers could be decreasing: we only get 3-2-1. So maybe the pattern is 3-2-1-3-2-1... Or maybe something else entirely! It's not really clear (at least to me) – Helen Jan 1 at 23:15
• Without looking at the source,this could also be a linear recurrence $a_{n+3}=2 a_n - a_{n+1}$. Then the next number would be $0$. – WimC Jan 2 at 8:40
• It seems the answer is not given in your book. This is not a book dedicated to puzzles but to data communication and information security. This puzzle must be related to the chapter it belongs to. What is that chapter about? – xhienne Feb 1 at 20:51

For $$n \geq 3$$, the numbers seem to follow the recursive relationship $$a_{n} = 16 - a_{n-1} - 2a_{n-2}$$ In particular, $$8 = 16 - 4 - (2\times 2)$$ $$0 = 16 - 8 - (2\times 4)$$ $$0 = 16 - 0 - (2\times 8)$$ $$16 = 16 - 0 - (2\times 0)$$ and finally $$0 = 16 - 16 - (2\times 0)$$