# $a_n = 2n^2, n \geq 1$ (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? Jan 1, 2021 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) Jan 1, 2021 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, 2021 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? Feb 1, 2021 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 $$0 = 16 - 16 - (2\times 0)$$