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

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Answer 1 · 2015-07-31T15:53:15

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

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 $a_n = 2^n-n^2$