201220: [AtCoder]ARC122 A - Many Formulae
Description
Score : $400$ points
Problem Statement
You are given a sequence of $N$ non-negative integers: $A_1,A_2,\cdots,A_N$.
Consider inserting a + or - between each pair of adjacent terms to make one formula.
There are $2^{N-1}$ such ways to make a formula. Such a formula is called good when the following condition is satisfied:
-does not occur twice or more in a row.
Find the sum of the evaluations of all good formulae. We can prove that this sum is always a non-negative integer, so print it modulo $(10^9+7)$.
Constraints
- $1 \leq N \leq 10^5$
- $1 \leq A_i \leq 10^9$
- All values in input are integers.
Input
Input is given from Standard Input in the following format:
$N$ $A_1$ $A_2$ $\cdots$ $A_N$
Output
Print the sum modulo $(10^9+7)$.
Sample Input 1
3 3 1 5
Sample Output 1
15
We have the following three good formulae:
-
$3+1+5=9$
-
$3+1-5=-1$
-
$3-1+5=7$
Note that $3-1-5$ is not good since- occurs twice in a row in it.
Thus, the answer is $9+(-1)+7=15$.
Sample Input 2
4 1 1 1 1
Sample Output 2
10
We have the following five good formulae:
-
$1+1+1+1=4$
-
$1+1+1-1=2$
-
$1+1-1+1=2$
-
$1-1+1+1=2$
-
$1-1+1-1=0$
Thus, the answer is $4+2+2+2+0=10$.
Sample Input 3
10 866111664 178537096 844917655 218662351 383133839 231371336 353498483 865935868 472381277 579910117
Sample Output 3
279919144
Print the sum modulo $(10^9+7)$.