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 OCaml (4.02.3) (judge verdict: Accepted)

module M = Map.Make (struct type t = int let compare = compare end)

let () =
    let split s =
        let rec loop i cur flag acc =
            if i = String.length s then ( if flag then cur :: acc else acc) else (
                match flag, s.[i] with
                | true,  ' ' -> loop (i + 1) "" false (cur :: acc)
                | false, ' ' -> loop (i + 1) "" false acc
                | true,  v   -> loop (i + 1) (Printf.sprintf "%s%c" cur v) true acc
                | false, v   -> loop (i + 1) (Printf.sprintf "%s%c" cur v) true acc
            )
        in
        List.map int_of_string (List.rev (loop 0 "" false []))
    in
    let main () =
        let n, k = Scanf.sscanf (read_line ()) "%d %d" (fun n k -> n, k) in
        let a = Array.of_list (split (read_line ())) in
        let arr = Array.make (n + 1) 0 in
        let rec loop i sum =
            if i < n then (
                let sum = sum + a.(i) - 1 in
                arr.(i + 1) <- sum mod k;
                loop (i + 1) sum
            )
        in
        loop 0 0;
        let add k v map = M.add k (v + try M.find k map with _ -> 0) map in
        let rec loop i map acc =
            if i > n then acc else
                let map = if i >= k then add arr.(i - k) (-1) map else map in
                let acc = acc + try M.find arr.(i) map with _ -> 0 in
                let map = add arr.(i) 1 map in
                loop (i + 1) map acc
        in
        let r = loop 1 (M.add 0 1 M.empty) 0 in
        Printf.printf "%d\n" r
    in
    main ()