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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#ifndef LOCAL
#pragma GCC optimize ("O3")
#pragma GCC optimize ("unroll-loops")
#pragma GCC target ("avx")
#endif

#include <vector>
#include <set>
#include <map>
#include <tuple>
#include <queue>
#include <stack>
#include <bitset>
#include <functional>
#include <algorithm>
#include <utility>
#include <numeric>
#include <iostream>
#include <cmath>
#include <cassert>
#include <cstring>

using namespace std;

using ll = long long;
using VI = vector<int>;
using VVI = vector<vector<int>>;
using VLL = vector<ll>;
using VVLL = vector<vector<ll>>;
using VB = vector<bool>;
using PII = pair<int, int>;
using PLL = pair<ll, ll>;
template <typename T>
using minheap = priority_queue<T, vector<T>, greater<T>>;
const int INF = 1000000007;
const ll INF_LL = 1'000'000'000'000'000'007;

#define __overload3(_1, _2, _3, name,...) name
#define rep(...) __overload3(__VA_ARGS__, repFromUntil, repUntil, repeat)(__VA_ARGS__)
#define repeat(times) repFromUntil(_name##__LINE__, 0, times)
#define repUntil(name, times) repFromUntil(name, 0, times)
#define repFromUntil(name, from, until) for (int name = from, name##__until = (until); name < name##__until; name++)
#define repr(...) __overload3(__VA_ARGS__, reprFromUntil, reprUntil, repeat)(__VA_ARGS__)
#define reprUntil(name, times) reprFromUntil(name, 0, times)
#define reprFromUntil(name, from, until) for (int name = (until) - 1, name##__from = (from); name >= name##__from; name--)

#define EXIT(out) do { OUT(out); exit(0); } while (0)

#define all(x) begin(x), end(x)
#define rall(x) rbegin(x), rend(x)
#define fs first
#define sn second

#ifdef LOCAL
#define debug(v) do {debugos << "L" << __LINE__ << " " << #v << " > ";debugos<<(v)<<newl;} while (0)
#define debugv(v) do {debugos << "L" << __LINE__ << " " << #v << " > ";for(auto e:(v)){debugos<<e<<" ";}debugos<<newl;} while (0)
#define debuga(m,w) do {debugos << "L" << __LINE__ << " " << #m << " > ";for(int x=0;x<(w);x++){debugos<<(m)[x]<<" ";}debugos<<newl;} while (0)
#define debugaa(m,h,w) do {debugos << "L" << __LINE__ << " " << #m << " > \n";for(int y=0;y<(h);y++){for(int x=0;x<(w);x++){debugos<<(m)[y][x]<<" ";}debugos<<newl;}} while (0)
#else
#define debug(v) do {v;} while (0)
#define debugv(v) do; while (0)
#define debuga(m,w) do; while (0)
#define debugaa(m,h,w) do; while (0)
#endif

#define newl "\n"
constexpr int dr[] = {1,-1,0,0};  // LRUD
constexpr int dc[] = {0,0,1,-1};

bool inside(int r, int c, int H, int W) {
  return 0 <= r and r < H and 0 <= c and c < W;
}

template <typename T, typename U> bool chmin(T& var, U x) {
  if (var > x) {
    var = x;
    return true;
  } else return false;
}

template <typename T, typename U> bool chmax(T& var, U x) {
  if (var < x) {
    var = x;
    return true;
  } else return false;
}

template <typename T> int sgn(T val) { return (T(0) < val) - (val < T(0)); }

ll power(ll e, int t, ll mod = INF_LL) {
  ll res = 1;
  while (t) {
    if (t&1) res = (res * e) % mod;
    t >>= 1;
    e = (e * e) % mod;
  }
  return res;
}

template <typename T> T divceil(T m, T d) {
  assert(m >= 0 and d > 0);
  return (m+d-1)/d;
}

template<typename T>
vector<T> make_v(size_t a, T b) { return vector<T>(a, b); }

template<typename... Ts>
auto make_v(size_t a, Ts... ts) {
  return vector<decltype(make_v(ts...))>(a, make_v(ts...));
}

string operator*(const string& s, int times) {
  string res = "";
  rep(times) res += s;
  return res;
}

class MyScanner {
public:
  int offset = 0;
  char nc(){
#ifdef LOCAL
    return getchar();
#else
    static char buf[100000],*L=buf,*R=buf;
    return L==R&&(R=(L=buf)+fread(buf,1,100000,stdin),L==R)?EOF:*L++;
#endif
  }
  template <typename T> void input_integer(T& var) {
    var = 0; T sign = 1;
    int cc = nc();
    for (; cc<'0' || '9'<cc; cc = nc())
      if (cc == '-') sign = -1;
    for (; '0' <= cc && cc <= '9'; cc = nc())
      var = (var << 3) + (var << 1) + cc - '0';
    var = var * sign; var += offset;
  }
  int c() { char c; while (c = nc(), c == ' ' or c == '\n'); return c; }
  MyScanner& operator>>(char& var) { var = c(); return *this; }
  MyScanner& operator>>(int& var) { input_integer<int>(var); return *this; }
  MyScanner& operator>>(ll& var) { input_integer<ll>(var); return *this; }
  MyScanner& operator>>(string& var) {
    var = ""; int cc = nc();
    for (; !isvisiblechar(cc); cc = nc());
    for (; isvisiblechar(cc); cc = nc())
      var.push_back(cc);
    return *this;
  }
  template <typename T>
  operator T() { T x; *this >> x; return x; }
  template <typename T>
  void operator()(T &t) { *this >> t; }
  template <typename T, typename... Ts>
  void operator()(T &t, Ts &...ts) { *this >> t; this->operator()(ts...); }
  template <typename Iter>
  void iter(Iter first, Iter last) { while (first != last) *this >> *first, first++; }
  VI vi(int n) { VI res(n); iter(all(res)); return res; }
  VVI vvi(int n, int m) { VVI res(n); rep(i, n) res[i] = vi(m); return res; }
  VLL vll(int n) { VLL res(n); iter(all(res)); return res; }
  VVLL vvll(int n, int m) { VVLL res(n); rep(i, n) res[i] = vll(m); return res; }
  template <typename T>
  vector<T> v(int n) { vector<T> res(n); iter(all(res)); return res; }
private:
  int isvisiblechar(int c) { return 0x21 <= c && c <= 0x7E; }
} IN, IN1{-1};

class MyPrinter {
public:
  int offset = 0;
  template <typename T>
  void output_integer(T var) {
    var += offset;
    if (var == 0) { putchar('0'); return; }
    if (var < 0) putchar('-'), var = -var;
    char stack[32]; int stack_p = 0;
    while (var) stack[stack_p++] = '0' + (var % 10), var /= 10;
    while (stack_p) putchar(stack[--stack_p]);
  }
  MyPrinter& operator<<(char c) { putchar(c); return *this; }
  MyPrinter& operator<<(double x) { printf("%.10f", x); return *this; }
  template <typename T> MyPrinter& operator<<(T var) { output_integer<T>(var); return *this; }
  MyPrinter& operator<<(char* str_p) { while (*str_p) putchar(*(str_p++)); return *this; }
  MyPrinter& operator<<(const char* str_p) { while (*str_p) putchar(*(str_p++)); return *this; }
  MyPrinter& operator<<(const string& str) {
    const char* p = str.c_str();
    const char* l = p + str.size();
    while (p < l) putchar(*p++);
    return *this;
  }
  template <typename T>
  void operator()(T x) { *this << x << newl; }
  template <typename T, typename... Ts>
  void operator()(T x, Ts ...xs) { *this << x << " "; this->operator()(xs...); }
  template <typename Iter>
  void iter(Iter s, Iter t) {
    if (s == t) *this << "\n";
    else for (; s != t; s++) *this << *s << " \n"[next(s, 1) == t];
  }
  template <typename Range>
  void range(const Range& r) { iter(begin(r), end(r)); }
} OUT, OUT1{1};

class DebugPrint {
public:
  template <typename T>
  DebugPrint& operator <<(const T& v) {
#ifdef LOCAL
    cerr << v;
#endif
    return *this;
  }
} debugos;

template <typename OutStream, typename T, typename U>
OutStream& operator<<(OutStream& out, const pair<T, U>& var) {
  return out << var.first << " " << var.second;
}
template <typename OutStream, typename Tuple, size_t I, size_t N,
          enable_if_t<I == N>* = nullptr>
OutStream& tuple_impl(OutStream& out, const Tuple& var) {
  return out;
}
template <typename OutStream, typename Tuple, size_t I, size_t N,
          enable_if_t<I != N>* = nullptr>
OutStream& tuple_impl(OutStream& out, const Tuple& var) {
  out << get<I>(var) << " ";
  return tuple_impl<OutStream, Tuple, I+1, N>(out, var);
}
template <typename OutStream, typename... Ts>
OutStream& operator<<(OutStream& out, const tuple<Ts...>& var) {
  return tuple_impl<OutStream, tuple<Ts...>, 0, sizeof...(Ts)>(out, var);
}
template <typename InStream, typename T, typename U>
InStream& operator>>(InStream& in, pair<T, U>& var) {
  return in >> var.first >> var.second;
}
template <typename InStream, typename Tuple, size_t I, size_t N,
          enable_if_t<I == N>* = nullptr>
InStream& tuple_impl(InStream& in, Tuple& var) {
  return in;
}
template <typename InStream, typename Tuple, size_t I, size_t N,
          enable_if_t<I != N>* = nullptr>
InStream& tuple_impl(InStream& in, Tuple& var) {
  in >> get<I>(var);
  return tuple_impl<InStream, Tuple, I+1, N>(in, var);
}
template <typename InStream, typename... Ts>
InStream& operator>>(InStream& in, tuple<Ts...>& var) {
  return tuple_impl<InStream, tuple<Ts...>, 0, sizeof...(Ts)>(in, var);
}


int main() {
  ll n = IN, k = IN;
  auto a = IN.vll(n);
  rep(i, n) a[i] = (a[i]+k-1)%k;
  VLL acc(a.size()+1); rep(i, a.size()) acc[i+1] = (acc[i] + a[i])%k;
  map<ll, int> cnt;
  ll res = 0;
  rep(i, n+1) {
    if (i-k>=0) cnt[acc[i-k]]--;
    res += cnt[acc[i]]++;
  }
  OUT(res);
}