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

Jul 31, 2015

${a}_{n} = {2}^{n} - {n}^{2}$ for $n = 0 , 1 , 2 , \ldots$

#### Explanation:

$1 , 1 , 0 , - 1 , 0 , 7 , 28 , 79 , 192$

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 ${a}_{n} = {2}^{n} - {n}^{2}$