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 C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;

namespace abc140_e
{
    class Program
    {
        private static readonly Dictionary<Tuple<int, int>, Tuple<int, int>> secondMemo =
            new Dictionary<Tuple<int, int>, Tuple<int, int>>();

        static void Main(string[] args)
        {
            var n = int.Parse(Console.ReadLine());
            var pList = Console.ReadLine().Split(' ').Select(int.Parse).ToList();


            var sum = 0;
            foreach (var rightAt0 in GetRange(1, n))
            {
                foreach (var l in GetRange(0, n - rightAt0))
                {
                    Tuple<int, int> secondMax;
                    var r = l + rightAt0;
                    if (rightAt0 == 1)
                    {
                        var lValue = pList[l];
                        var rValue = pList[r];
                        secondMax = lValue < rValue // (second, first)
                            ? new Tuple<int, int>(lValue, rValue)
                            : new Tuple<int, int>(rValue, lValue);
                    }
                    else
                    {
                        var memo = GetMemo(l, r - 1);
                        secondMax = GetSecondMax(memo.Item1, memo.Item2, pList[r]);
                    }

                    sum += secondMax.Item1;
                    Memoization(l, r, secondMax);
                }
            }

            Console.WriteLine(sum);
        }

        private static void Memoization(int l, int r, Tuple<int, int> value)
        {
            secondMemo[new Tuple<int, int>(l, r)] = value;
        }

        private static Tuple<int, int> GetMemo(int l, int r)
        {
            return secondMemo[new Tuple<int, int>(l, r)];
        }

        private static Tuple<int, int> GetSecondMax(int second, int max, int newValue)
        {
            if (second > max)
            {
                throw new ArgumentException("second > max");
            }

            if (newValue < second)
            {
                return new Tuple<int, int>(second, max);
            }

            if (max < newValue)
            {
                return new Tuple<int, int>(max, newValue);
            }

            return new Tuple<int, int>(newValue, max);
        }

        private static IEnumerable<int> GetRange(int from, int until)
        {
            return Enumerable.Range(from, until - from);
        }
    }
}