Problem:
Score : 500 points

Problem Statement

Given are a sequence of N positive integers A_1, A_2, \ldots, A_N, and a positive integer K.

Find the number of non-empty contiguous subsequences in A such that the remainder when dividing the sum of its elements by K is equal to the number of its elements. We consider two subsequences different if they are taken from different positions, even if they are equal sequences.

Constraints

All values in input are integers.

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

1 \leq K \leq 10^9

1 \leq A_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N K
A_1 A_2 \cdots A_N

Output

Print the number of subsequences that satisfy the condition.

Sample Input 1

5 4
1 4 2 3 5

Sample Output 1

4

Four sequences satisfy the condition: (1), (4,2), (1,4,2), and (5).

Sample Input 2

8 4
4 2 4 2 4 2 4 2

Sample Output 2

7

(4,2) is counted four times, and (2,4) is counted three times.

Sample Input 3

10 7
14 15 92 65 35 89 79 32 38 46

Sample Output 3

8

Solution in Rust (1.15.1) (judge verdict: Time Limit Exceeded)

use std::io::*;
use utils::*;

pub fn main() {
    let i = stdin();
    let mut o = Vec::new();
    run(i.lock(), &mut o);
    stdout().write_all(&o).unwrap();
}

pub fn run<R: BufRead, W: Write>(i: R, o: &mut W) {
    let mut i = CpReader::new(i);
    let (n, k) = i.read::<(usize, u64)>();
    let mut sums = Vec::with_capacity(n + 1);
    sums.push(0);
    let mut sum = 0;
    for a in i.read_iter::<u64>(n) {
        sum += a;
        sums.push(sum);
    }
    let mut ans = 0;
    for l in 1..n + 1 {
        for i in 0..sums.len() - l {
            if (sums[i + l] - sums[i]) % k == l as u64 {
                ans += 1;
            }
        }
    }
    writeln!(o, "{}", ans).unwrap();
}

pub mod utils {
    use super::*;

    pub struct CpReader<R: BufRead> {
        r: R,
        b: Vec<u8>,
    }
    impl<R: BufRead> CpReader<R> {
        pub fn new(r: R) -> Self {
            CpReader {
                r: r,
                b: Vec::new(),
            }
        }
        pub fn read_word(&mut self) -> &[u8] {
            self.b.clear();
            let mut consume = 0;
            loop {
                self.r.consume(consume);
                let b = self.r.fill_buf().unwrap();
                assert!(b.len() != 0);
                if let Some(p) = b.iter().position(|&x| x == b' ' || x == b'\n') {
                    self.b.extend_from_slice(&b[..p]);
                    consume = p + 1;
                    break;
                }
                self.b.extend_from_slice(b);
                consume = b.len();
            }
            self.r.consume(consume);
            &self.b
        }
        pub fn read_word_str(&mut self) -> &str {
            unsafe { std::str::from_utf8_unchecked(self.read_word()) }
        }

        pub fn read_line(&mut self) -> &[u8] {
            self.b.clear();
            self.r.read_until(b'\n', &mut self.b).unwrap();
            if self.b.last() == Some(&b'\n') {
                &self.b[..self.b.len() - 1]
            } else {
                &self.b
            }
        }
        pub fn read_line_str(&mut self) -> &str {
            unsafe { std::str::from_utf8_unchecked(self.read_line()) }
        }

        pub fn read<T: CpIn>(&mut self) -> T {
            T::read_from(self)
        }

        pub fn read_vec<T: CpIn>(&mut self, n: usize) -> Vec<T> {
            (0..n).map(|_| self.read()).collect()
        }
        pub fn read_iter<'a, T: CpIn>(&'a mut self, n: usize) -> CpIter<'a, R, T> {
            CpIter {
                r: self,
                n: n,
                _pd: Default::default(),
            }
        }
    }
    pub struct CpIter<'a, R: BufRead + 'a, T> {
        r: &'a mut CpReader<R>,
        n: usize,
        _pd: std::marker::PhantomData<fn() -> T>,
    }
    impl<'a, R: BufRead, T: CpIn> Iterator for CpIter<'a, R, T> {
        type Item = T;
        fn next(&mut self) -> Option<T> {
            if self.n == 0 {
                None
            } else {
                self.n -= 1;
                Some(self.r.read())
            }
        }
    }

    pub trait CpIn {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self;
    }

    impl CpIn for u64 {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
            read_u64_fast(&mut r.r)
        }
    }
    impl CpIn for i64 {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
            read_i64_fast(&mut r.r)
        }
    }
    impl CpIn for char {
        fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
            let b = r.r.fill_buf().unwrap()[0] as char;
            r.r.consume(1);
            let s = r.r.fill_buf().unwrap()[0];
            assert!(s == b' ' || s == b'\n');
            r.r.consume(1);
            b
        }
    }

    macro_rules! cpin_tuple {
        ($($t:ident),*) => {
            impl<$($t: CpIn),*> CpIn for ($($t),*) {
                fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
                    ($($t::read_from(r)),*)
                }
            }
        };
    }

    macro_rules! cpin_cast {
        ($t_self:ty, $t_read:ty) => {
            impl CpIn for $t_self {
                fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
                    <$t_read>::read_from(r) as $t_self
                }
            }
        };
    }
    macro_rules! cpin_parse {
        ($t:ty) => {
            impl CpIn for $t {
                fn read_from<R: BufRead>(r: &mut CpReader<R>) -> Self {
                    r.read_word_str().parse().unwrap()
                }
            }
        };
    }
    cpin_cast!(usize, u64);
    cpin_cast!(u32, u64);
    cpin_cast!(u16, u64);
    cpin_cast!(i32, i64);
    cpin_cast!(i16, i64);
    cpin_cast!(i8, i64);

    cpin_parse!(f64);

    cpin_tuple!(T1, T2);
    cpin_tuple!(T1, T2, T3);
    cpin_tuple!(T1, T2, T3, T4);
    cpin_tuple!(T1, T2, T3, T4, T5);

    fn read_u64_fast<R: BufRead>(r: &mut R) -> u64 {
        let mut value = 0;
        loop {
            let mut idx = 0;
            let buf_len;
            {
                let buf = r.fill_buf().unwrap();
                buf_len = buf.len();
                assert!(buf_len != 0);
                while idx < buf_len && b'0' <= buf[idx] && buf[idx] <= b'9' {
                    value = value * 10 + (buf[idx] - b'0') as u64;
                    idx += 1;
                }
            }
            if idx < buf_len {
                r.consume(idx + 1);
                return value;
            }
            r.consume(idx);
        }
    }
    fn read_i64_fast<R: BufRead>(r: &mut R) -> i64 {
        let (consume, sign) = match r.fill_buf().unwrap()[0] {
            b'+' => (true, 1),
            b'-' => (true, -1),
            _ => (false, 1),
        };
        if consume {
            r.consume(1);
        }
        read_u64_fast(r) as i64 * sign
    }
}