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: Time Limit Exceeded)

def maxi(l, r, pre = nil)
  return [P[l],P[l+1]].max if r - l == 1 
  pre ||= maxi(l,r-1)
  P[r] > maxi(l,r-1) ? P[r] : maxi(l,r-1)
end

def s_maxi(l, r, s_pre = nil, pre = nil)
  return [P[l],P[l+1]].min if r - l == 1 
  s_pre ||= s_maxi(l,r-1)
  pre ||= maxi(l,r-1)
  if s_pre > P[r]
    s_pre
  elsif s_pre < P[r] && P[r] < pre
    P[r]
  else
    pre
  end
end

N=gets.chomp.to_i
P=gets.chomp.split.map(&:to_i)
sum = 0

(N-1).times do |l|
  pre_s_maxi = nil
  pre_maxi = nil
  ((l+1)..(N-1)).each do |r|
    sum += s_maxi(l, r, pre_s_maxi, pre_maxi)
    pre_s_maxi = s_maxi(l,r,pre_s_maxi,pre_maxi)
    pre_maxi = maxi(l,r,pre_maxi)
  end
end

puts sum