Question

What is the pattern in the sequence 1, 1, 0, -1, 0, 7, 28, 79, 192?

Answer 1

`a_n = 2^n-n^2` for `n=0, 1, 2,...`

Explanation:

Start with the sequence:
`1, 1, 0, -1, 0, 7, 28, 79, 192`

Form the sequence of differences:
`0, -1, -1, 1, 7, 21, 52, 113`

Then the sequence of differences of those differences:
`-1, 0, 2, 6, 14, 30, 62`

Then the sequence of differences of those differences:
`1, 2, 4, 8, 16, 32`

Subtract this from the original sequence to get:
`0, -1, -4, -9, -16, -25`

This is obviously a sequence of negated square numbers.

So `a_n = 2^n-n^2`