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: Runtime Error)

import java.io.BufferedReader;
import java.io.Closeable;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.math.BigDecimal;
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collections;
import java.util.Comparator;
import java.util.HashMap;
import java.util.HashSet;
import java.util.Iterator;
import java.util.LinkedList;
import java.util.ListIterator;
import java.util.Map.Entry;

import java.util.PriorityQueue;
import java.util.Random;
import java.util.StringTokenizer;
import java.util.TreeMap;
import java.util.TreeSet;


public class Main {
	
	public static void main(String[] args) throws IOException {	
		try(final Scanner sc = new Scanner(System.in)){
			final int N = sc.nextInt();
			final long K = sc.nextLong();
			
			long[] as = new long[N];
			for(int i = 0; i < N; i++) {
				as[i] = sc.nextLong() % K;
			}
			
			long[] ts = new long[N];
			for(int i = 0; i < N; i++) {
				ts[i] = as[i];
			}
			
			for(int i = 1; i < N; i++) {
				ts[i] += ts[i - 1];
				ts[i] %= K;
			}
			
			for(int i = 0; i < N; i++) {
				ts[i] = (ts[i] - ((i + 1) % K) + K) % K;
			}
			
			/*
			System.out.println(Arrays.toString(as));
			System.out.println(Arrays.toString(ss));
			System.out.println(Arrays.toString(ts));
			*/
			
			HashMap<Long, Integer> counts = new HashMap<Long, Integer>();
			for(int i = 0; i < Math.min(N, K - 1); i++) {
				if(!counts.containsKey(ts[i])) { counts.put(ts[i], 0); }
				counts.put(ts[i], counts.get(ts[i]) + 1);
			}
			
			long answer = 0;
			/*
			for(int i = 0; i < N; i++) {
				long sum = 0;
				for(int j = i; j < N; j++) {
					sum += as[j];
					if(sum % K == (j - i + 1)) {
						System.out.println(i + " " + j + " " + (j - i + 1));
					}
				}
			}
			*/
			
			long X = 0;
			for(int i = 0; i < N; i++) {
				//final long X = (i == 0 ? 0 : (K + (as[i - 1] % K) - (i + 1))) % K;
				X += i == 0 ? 0 : ((K + as[i - 1] - 1) % K);
				X %= K;
				
				//System.out.println(X + " " + counts + " " + counts.get(X));
				if(counts.containsKey(X)) {
					answer += counts.get(X);
				}
				
				counts.put(ts[i], counts.get(ts[i]) - 1);

				if(i + (K - 1) < N) {
					final int index = (int)(i + (K - 1));
					if(!counts.containsKey(ts[index])) {
						counts.put(ts[index], 0);
					}
					
					counts.put(ts[index], counts.get(ts[index]) + 1);
				}
			}
			
			System.out.println(answer);
		}
	}

	public static class Scanner implements Closeable {
		private BufferedReader br;
		private StringTokenizer tok;

		public Scanner(InputStream is) throws IOException {
			br = new BufferedReader(new InputStreamReader(is));
		}

		private void getLine() throws IOException {
			while(!hasNext()){
				tok = new StringTokenizer(br.readLine());
			}
		}

		private boolean hasNext() {
			return tok != null && tok.hasMoreTokens();
		}

		public String next() throws IOException {
			getLine();
			return tok.nextToken();
		}

		public int nextInt() throws IOException {
			return Integer.parseInt(next());
		}

		public long nextLong() throws IOException {
			return Long.parseLong(next());
		}
		

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

		public int[] nextIntArray(int n) throws IOException {
			final int[] ret = new int[n];
			for(int i = 0; i < n; i++){
				ret[i] = this.nextInt();
			}
			return ret;
		}

		public long[] nextLongArray(int n) throws IOException {
			final long[] ret = new long[n];
			for(int i = 0; i < n; i++){
				ret[i] = this.nextLong();
			}
			return ret;
		}

		public void close() throws IOException {
			br.close();
		}
	}
}