Non-adjacent subsets

Find the number of subsets of $n$ such that if $i$ does not belong in the subset then either $i+1$ or $i-1$ belongs to the subset. (Hint: use double recursion, use fibonacci series)

Supersum

For a natural number $n$, the supersum of $n$ is defined as the sum of possible combinations that we can obtain by deleting digits of $n$, calculated as modulo 9. For example, the supersum of 1234 is:

No deletions: 1234 +
Deleting 1 digit: 123 + 134 + 234 + 124
Deleting 2 digits: 12 + 13 + 14 + 23 + 24 + 34 +
Deleting 3 digits: 1 + 2 + 3 + 4

$ = 8 \mod 9$

If $n$ is a 2011-digit number, and $s$ is its supersum, show that $s-n$ is divisible by 9.