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 Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.ArrayList;
import java.util.List;
import java.util.NoSuchElementException;

public class Main {

	public static void main(String[] args) {

		FastScanner sc = new FastScanner();
		PrintWriter out = new PrintWriter(System.out);

		long N=sc.nextLong();
		long K=sc.nextLong();
		List<Long> As=sc.nextArrayLongsN((int)N);
		int ans=0;
		int hidari=0;
		int kosuu=1;
		
		for(int i=0;i<N;i++) {
			As.set(i, As.get(i)%K);
		}
		long sumold=0;
		for(kosuu=1;kosuu<=N;kosuu++) {
			sumold+=As.get(kosuu-1);
			long sum=sumold;
			if(sum%K==kosuu) {
				ans++;
			}
			for(hidari=1;hidari+kosuu<=N;hidari++) {
				sum=sum-As.get(hidari-1)+As.get(hidari+kosuu-1);
				if(sum%K==kosuu) {
					ans++;
				}
			}
		}
		
		
		out.print(ans);
		out.flush();

	}


}

class FastScanner {
	private final InputStream in = System.in;
	private final byte[] buffer = new byte[1024];
	private int ptr = 0;
	private int buflen = 0;

	private boolean hasNextByte() {
		if (ptr < buflen) {
			return true;
		} else {
			ptr = 0;
			try {
				buflen = in.read(buffer);
			} catch (IOException e) {
				e.printStackTrace();
			}
			if (buflen <= 0) {
				return false;
			}
		}
		return true;
	}

	private int readByte() {
		if (hasNextByte())
			return buffer[ptr++];
		else
			return -1;
	}

	private static boolean isPrintableChar(int c) {
		return 33 <= c && c <= 126;
	}

	public boolean hasNext() {
		while (hasNextByte() && !isPrintableChar(buffer[ptr]))
			ptr++;
		return hasNextByte();
	}

	public String next() {
		if (!hasNext())
			throw new NoSuchElementException();
		StringBuilder sb = new StringBuilder();
		int b = readByte();
		while (isPrintableChar(b)) {
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}

	public long nextLong() {
		if (!hasNext())
			throw new NoSuchElementException();
		long n = 0;
		boolean minus = false;
		int b = readByte();
		if (b == '-') {
			minus = true;
			b = readByte();
		}
		if (b < '0' || '9' < b) {
			throw new NumberFormatException();
		}
		while (true) {
			if ('0' <= b && b <= '9') {
				n *= 10;
				n += b - '0';
			} else if (b == -1 || !isPrintableChar(b)) {
				return minus ? -n : n;
			} else {
				throw new NumberFormatException();
			}
			b = readByte();
		}
	}

	public int nextInt() {
		long nl = nextLong();
		if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE)
			throw new NumberFormatException();
		return (int) nl;
	}

	public double nextDouble() {
		return Double.parseDouble(next());
	}

	public int[] nextIntsN(int N) {
		int[] ints = new int[N];
		for (int i = 0; i < N; i++) {
			ints[i] = nextInt();
		}
		return ints;
	}

	public long[] nextLongsN(int N) {
		long[] longs = new long[N];
		for (int i = 0; i < N; i++) {
			longs[i] = nextLong();
		}
		return longs;
	}

	public char[] nextCharsN(int N) {
		char[] chars = new char[N];
		String msg = next();
		for (int i = 0; i < N; i++) {
			chars[i] = msg.charAt(i);
		}
		return chars;
	}

	public char[][] nextCharsHW(int H, int W) {
		char[][] chars = new char[H][W];
		for (int i = 0; i < H; i++) {
			String str = next();
			for (int j = 0; j < W; j++) {
				chars[i][j] = str.charAt(j);
			}
		}
		return chars;
	}

	public int[][] nextIntsNM(int N, int M) {
		int[][] ints = new int[N][M];
		for (int i = 0; i < N; i++) {
			for (int j = 0; j < M; j++) {
				ints[i][j] = nextInt();
			}
		}
		return ints;
	}

	public List<Integer> nextArrayIntsN(int N) {
		ArrayList<Integer> array = new ArrayList<>(N);
		for (int i = 0; i < N; i++) {
			array.add(nextInt());
		}
		return array;
	}

	public List<Long> nextArrayLongsN(int N) {
		ArrayList<Long> array = new ArrayList<>(N);
		for (int i = 0; i < N; i++) {
			array.add(nextLong());
		}
		return array;
	}

	public List<List<Long>> nextArrayLongsNM(int N, int M) {

		List<List<Long>> arrays = new ArrayList<List<Long>>(M);
		for (int i = 0; i < M; i++) {
			List<Long> array = nextArrayLongsN(N);
			arrays.add(array);
		}
		return arrays;
	}

}