Problem:
Score : 300 points

Problem Statement

Snuke has an integer sequence A of length N.

He will freely choose an integer b.
Here, he will get sad if A_i and b+i are far from each other.
More specifically, the sadness of Snuke is calculated as follows:

abs(A_1 - (b+1)) + abs(A_2 - (b+2)) + ... + abs(A_N - (b+N))

Here, abs(x) is a function that returns the absolute value of x.

Find the minimum possible sadness of Snuke.

Constraints

1 \leq N \leq 2 \times 10^5

1 \leq A_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 ... A_N

Output

Print the minimum possible sadness of Snuke.

Sample Input 1

5
2 2 3 5 5

Sample Output 1

2

If we choose b=0, the sadness of Snuke would be abs(2-(0+1))+abs(2-(0+2))+abs(3-(0+3))+abs(5-(0+4))+abs(5-(0+5))=2.
Any choice of b does not make the sadness of Snuke less than 2, so the answer is 2.

Sample Input 2

9
1 2 3 4 5 6 7 8 9

Sample Output 2

0

Sample Input 3

6
6 5 4 3 2 1

Sample Output 3

18

Sample Input 4

7
1 1 1 1 2 3 4

Sample Output 4

6

Solution in Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

import qualified Data.List as L
import qualified Data.Map as Map

getInts :: IO [Integer]
getInts = map read . words <$> getLine

main :: IO()
main = do
  n <- getLine
  a <- getInts
  let aList = zipWith (-) a [1..]
  putStrLn $ show $ slv aList

slv :: [Integer] -> Integer
slv aList = minimum [slv0, slv1, slv2, slv3, slv4, slv5, slv6, slv7]
  where slv0 = sum $ map (\x -> abs(x - (roundUpOn5 (average aList)))) aList
        slv1 = sum $ map (\x -> abs(x - 1 - (roundUpOn5 (average aList)))) aList
        slv2 = sum $ map (\x -> abs(x + 1 - (roundUpOn5 (average aList)))) aList
        slv3 = sum $ map (\x -> abs(x - maximum aList)) aList
        slv4 = sum $ map (\x -> abs(x - minimum aList)) aList
        slv5 = sum $ map (\x -> abs(x - (roundUpOn5 (var aList)))) aList
        slv6 = sum $ map (\x -> abs(x - 1 - (roundUpOn5 (var aList)))) aList
        slv7 = sum $ map (\x -> abs(x + 1 - (roundUpOn5 (var aList)))) aList

var :: [Integer] -> Double
var a = aveA2 - (aveA * aveA)
  where aveA = average a
        aveA2 = average2 a

average :: [Integer] -> Double 
average a = (realToFrac (sum a)) / (realToFrac (length a))

average2 :: [Integer] -> Double
average2 a = (realToFrac (sum [x*x | x <- a])) / (realToFrac (length a))

roundUpOn5 :: (RealFrac a, Integral b) => a -> b
roundUpOn5 x
  | n <= -0.5 = m - 1
  | n >= 0.5 = m + 1
  | otherwise = m
  where (m, n) = properFraction x