CodeForces - 447E DZY Loves Fibonacci Numbers （线段树）

题目链接：http://codeforces.com/contest/447/problem/E
题意：在一个序列中有两种操作：第一种是修改操作，在某个区间加上斐波那契数列的一段；第二种是查询操作，查询某个区间的区间和。
E. DZY Loves Fibonacci Numbers
time limit per test4 seconds
memory limit per test256 megabytes
inputstandard input
outputstandard output

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

F1 = 1; F2 = 1; Fn = Fn - 1 + Fn - 2 (n > 2).
DZY loves Fibonacci numbers very much. Today DZY gives you an array consisting of n integers: a1, a2, …, an. Moreover, there are m queries, each query has one of the two types:

Format of the query “1 l r”. In reply to the query, you need to add Fi - l + 1 to each element ai, where l ≤ i ≤ r.
Format of the query “2 l r”. In reply to the query you should output the value of modulo 1000000009 (109 + 9).
Help DZY reply to all the queries.

Input
The first line of the input contains two integers n and m (1 ≤ n, m ≤ 300000). The second line contains n integers a1, a2, …, an (1 ≤ ai ≤ 109) — initial array a.

Then, m lines follow. A single line describes a single query in the format given in the statement. It is guaranteed that for each query inequality 1 ≤ l ≤ r ≤ n holds.

Output
For each query of the second type, print the value of the sum on a single line.

Examples
input
4 4
1 2 3 4
1 1 4
2 1 4
1 2 4
2 1 3
output
17
12
Note
After the first query, a = [2, 3, 5, 7].

For the second query, sum = 2 + 3 + 5 + 7 = 17.

After the third query, a = [2, 4, 6, 9].

For the fourth query, sum = 2 + 4 + 6 = 12.

解题思路：
打表找出斐波那契数列的前$n$项和的规律：
$f_n+f_{n-1}+f_{(}n-2)+...+f_1=f_{n+2}-f_2$
（其实这个规律适用于相邻两项之和等于第三项的序列）

具体做法：
在每个线段树节点存3个值：分别为$ta$