Problem:
Score: 500 points

Problem Statement

Given is a permutation P of \{1, 2, \ldots, N\}.

For a pair (L, R) (1 \le L \lt R \le N), let X_{L, R} be the second largest value among P_L, P_{L+1}, \ldots, P_R.

Find \displaystyle \sum_{L=1}^{N-1} \sum_{R=L+1}^{N} X_{L,R}.

Constraints

2 \le N \le 10^5

1 \le P_i \le N

P_i \neq P_j (i \neq j)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
P_1 P_2 \ldots P_N

Output

Print \displaystyle \sum_{L=1}^{N-1} \sum_{R=L+1}^{N} X_{L,R}.

Sample Input 1

3
2 3 1

Sample Output 1

5

X_{1, 2} = 2, X_{1, 3} = 2, and X_{2, 3} = 1, so the sum is 2 + 2 + 1 = 5.

Sample Input 2

5
1 2 3 4 5

Sample Output 2

30

Sample Input 3

8
8 2 7 3 4 5 6 1

Sample Output 3

136

Solution in PyPy3 (2.4.0) (judge verdict: Accepted)

from bisect import*
class SqrtSet:
    def __init__(self,block_limit=201):
        self.key=[]
        self.child=[[]]
        self.block_limit=block_limit
    def le(self,key):
        ret=key
        i=bisect_left(self.key,key)
        if i:ret=self.key[i-1]
        block=self.child[i]
        i=bisect_left(block,key)
        if i:ret=block[i-1]
        return ret
    def ge(self,key):
        ret=key
        i=bisect(self.key,key)
        if i<len(self.key):ret=self.key[i]
        block=self.child[i]
        i=bisect(block,key)
        if i<len(block):ret=block[i]
        return ret
    def insert(self,key):
        i=bisect(self.key,key)
        block=self.child[i]
        insort(block,key)
        if len(block)==self.block_limit:
            sep=self.block_limit//2
            self.key.insert(i,block[sep])
            self.child.insert(i+1,block[sep+1:])
            self.child[i]=block[:sep]
def main():
    n,*p=map(int,open(0).read().split())
    l=[0]*n
    for i,v in enumerate(p,1):l[v-1]=i
    t=SqrtSet()
    insert,le,ge=t.insert,t.le,t.ge
    insert(0),insert(n+1)
    a=0
    for i,v in list(enumerate(l,1))[::-1]:
        f,b=ge(v),le(v)
        g,c=ge(f),le(b)
        a+=i*((f-v)*(b-c)+(v-b)*(g-f))
        insert(v)
    print(a)
main()