Problem:
Score : 400 points

Problem Statement

We have N dice arranged in a line from left to right. The i-th die from the left shows p_i numbers from 1 to p_i with equal probability when thrown.

We will choose K adjacent dice, throw each of them independently, and compute the sum of the numbers shown. Find the maximum possible value of the expected value of this sum.

Constraints

1 ≤ K ≤ N ≤ 200000

1 ≤ p_i ≤ 1000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
p_1 ... p_N

Output

Print the maximum possible value of the expected value of the sum of the numbers shown.

Your output will be considered correct when its absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

5 3
1 2 2 4 5

Sample Output 1

7.000000000000

When we throw the third, fourth, and fifth dice from the left, the expected value of the sum of the numbers shown is 7. This is the maximum value we can achieve.

Sample Input 2

4 1
6 6 6 6

Sample Output 2

3.500000000000

Regardless of which die we choose, the expected value of the number shown is 3.5.

Sample Input 3

10 4
17 13 13 12 15 20 10 13 17 11

Sample Output 3

32.000000000000

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

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

class Program
{
    static void Main(string[] args)
    {
        var inputs = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();
        var n = inputs[0];
        var k = inputs[1];
        var pk = Console.ReadLine().Split(' ').Select(int.Parse).ToArray();

        var expectedValues = new double[n];

        // 期待値入れる
        for(int i = 0; i < n; i++)
        {
            expectedValues[i] = pk[i] * Enumerable.Range(1, pk[i] - 1).Sum();
        }

        // 累積和
        var cSum = new CumulativeSum(expectedValues);

        // 探す
        double answer = 0;
        for(int i = 0; i <= n - k; i++)
        {
            var sum = cSum.GetSum(i, i + k);
            if (answer < sum) answer = sum;
        }

        Console.WriteLine(answer);
    }
}

/// <summary>
/// 累積和
/// </summary>
public class CumulativeSum
{
    public double[] Array { get; }

    public CumulativeSum(float[] array) : this(array.Select(t => (double)t).ToArray()) { }

    public CumulativeSum(double[] array)
    {
        var length = array.Length;
        Array = new double[length + 1];

        for (int i = 0; i < length; i++)
        {
            Array[i + 1] = array[i] + Array[i];
        }
    }

    /// <summary>
    /// left以上right未満の総和
    /// </summary>
    /// <param name="left">以上</param>
    /// <param name="right">未満</param>
    public double GetSum(int left, int right)
    {
        if (left >= right) return 0;

        return Array[right] - Array[left];
    }
}

public static class Combination
{
    public static IEnumerable<T[]> Enumerate<T>(IEnumerable<T> items, int k, bool withRepetition)
    {
        if (k == 1)
        {
            foreach (var item in items)
                yield return new T[] { item };
            yield break;
        }
        foreach (var item in items)
        {
            var leftside = new T[] { item };

            // item よりも前のものを除く （順列と組み合わせの違い)
            // 重複を許さないので、unusedから item そのものも取り除く
            var unused = withRepetition ? items : items.SkipWhile(e => !e.Equals(item)).Skip(1).ToList();

            foreach (var rightside in Enumerate(unused, k - 1, withRepetition))
            {
                yield return leftside.Concat(rightside).ToArray();
            }
        }
    }
}