-- More recursion

plus1 10 = x + 10 -- first we define add something to x, and then initialize it
    where x = 12

plus11 = let x = 11 in x + 23 -- in addition to previous example where x = 12(so we have 12 + 11 = 23)

factorial1 0 = 0
factorial1 x = factorial1(x - 1) * x -- it doesn't matter if we first put (factorial1(x - 1)) or x

{-- Find m-th summation of first n natural numbers.
If m > 1
  SUM(n, m) = SUM(SUM(n, m - 1), 1)
Else
  SUM(n, 1) = Sum of first n natural numbers.
--}

sumOfNaturalNumbers 1 = 1
sumOfNaturalNumbers n = n + sumOfNaturalNumbers(n - 1)

sumOfNaturalNumbers1 x = fromIntegral (x * (x + 1) / 2) -- or there's another way for that - div(x * (x + 1) / 2)

summation n 1 = sumOfNaturalNumbers n
summation n m = summation (summation n (m - 1)) 1

{--
Write function calculateF
calculateF(n) = n, if n < 3
calculateF(n) = calculateF(n - 1) + 2*calculateF(n - 2) + calculateFf(n - 3), otherwise
--}

-- here is the first way to do it, with guards(it's more optimal solution)
calculateF n
    | n < 3 = n
    | otherwise = calculateF(n - 1) + 2*calculateF(n - 2) + calculateF(n - 3)

-- here is a way to do it with if statements, but it's better with guards

calculateF1 n = if n < 3 then n else calculateF1(n - 1) + 2*calculateF1(n - 2) + calculateF1(n - 3)

-- Write function coundDigits n a b ,calculating how many time there is
-- the digit n in range of Integers [a,b]

countDigitsNumber 0 _ = 0
countDigitsNumber a n
    | mod a 10 == n = 1 + countDigitsNumber(div a 10) n
    | otherwise = countDigitsNumber(div a 10) n

countDigits n a b = countDigitsHelper a b 0
    where
    countDigitsHelper a b sum
        | a == (b + 1) = sum
        | a > b = error "Not valid input"
        | otherwise = countDigitsHelper (a + 1) b ((countDigitsNumber a n) + sum)

-- Write function calculateSin n by the formula in week03-task.png

factorial 1 = 1
factorial x = factorial(x - 1) * x

calculateTeylor x n = ((-1) ** n * x ** (2*n+1)) / factorial(2*n+1)

calculateSin n x = calculateSinIter 1 n 0
    where
        calculateSinIter begin end sum
            | begin == (end + 1) = sum
            | otherwise = calculateSinIter (begin + 1) end (sum + calculateTeylor x n)


{--
Calculate the numbers of all n-digit strictly increasing numbers
--}
-- Write function sumOfPrimes finding the sum of all prime numbers in the interval [a, b]
-- Write function canbePrime n finding if there is a way by remove one of the digits in n the number became a prime.