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 Ruby (2.3.3) (judge verdict: Accepted)

N = gets.chomp.to_i
p = Array.new(N+1)
str = gets.split
for i in 0...N do
    a = str[i].to_i
    p[a] = i+1
end

q = Array.new(3,N+1)
q[0] = 0
q[1] = p[N]
answer = 0

for i in 2..N do
    a = p[-i]
    l = 1
    r = i
    while l != r
        c = (l+r)/2
        if a < q[c]
            r = c
        else
            l = c+1
        end
    end
    q.insert(l,a)

    b = N-i+1
    if q[l-1] == 0
        answer += b*(q[l]-q[l-1])*(q[l+2]-q[l+1])
    elsif q[l+1] > N
        answer += b*(q[l-1]-q[l-2])*(q[l+1]-q[l])
    else
        answer += b*((q[l-1]-q[l-2])*(q[l+1]-q[l])+(q[l]-q[l-1])*(q[l+2]-q[l+1]))
    end
end
puts answer