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++ (Clang 10.0.0) (judge verdict: Accepted)

#include <algorithm>
#include <iostream>
#include <iomanip>
#include <numeric>
#include <vector>
#include <string>
#include <bitset>
#include <queue>
#include <tuple>
#include <regex>
#include <cmath>
#include <map>
#include <cstring>

using namespace std;

template<typename T> bool chmax( T &a, const T b ) { if ( a <= b ) { a = b; return ( true ); } else { return ( false ); } }
template<typename T> bool chmin( T &a, const T b ) { if ( a >= b ) { a = b; return ( true ); } else { return ( false ); } }

using ll = long long;
using Pint = pair<int, int>;
using Pll  = pair<ll, ll>;

#define eb emplace_back
#define pb push_back
#define mp make_pair
#define mt make_tuple

#define F first
#define S second

#define rep( i, n ) for ( ll i = 0; i < (ll)( n ); ++i )
#define reps( i, n ) for ( ll i = 1; i <= (ll)( n ); ++i )
#define rrep( i, n ) for ( ll i = (ll)( ( n ) - 1 ); i >= 0; --i )
#define rreps( i, n ) for ( ll i = (ll)( ( n ) ); i > 0; --i )
#define arep( i, v ) for ( auto &&i : ( v ) )

#define ALL( c ) ( c ).begin(), ( c ).end()
#define RALL( c ) ( c ).rbegin(), ( c ).rend()
#define UNIQUE( c ) ( c ).erase( unique( ( c ).begin(), ( c ).end() ), ( c ).end() )

template<typename T = ll> constexpr T MAX = numeric_limits<T>::max();
template<typename T> T gcd( const T a, const T b ) { return ( b ? gcd( b, a % b ) : a ); }
template<typename T> T lcm( const T a, const T b ) { return ( a / gcd( a, b ) * b ); }

template<typename T = ll> struct UnionFind {
	vector<T> par;

	UnionFind( T n ) : par( n, -1 ) {}

	void init( T n ) { par.assign( n, -1 ); }
	T find( T x ) {
		if ( par[x] < 0 ) return ( x );
		else return ( par[x] = find( par[x] ) );
	}
	bool isSame( T x, T y ) {
		return ( find( x ) == find( y ) );
	}
	bool unite( T x, T y ) {
		x = find( x ); y = find( y );
		if ( x == y ) return ( false );
		if ( par[x] > par[y] ) swap( x, y );
		par[x] += par[y];
		par[y] = x;
		return ( true );
	}
	T size( T x ) { return ( -par[find( x )] ); }
};

template<typename T = ll> class CompareMax {
public:	T operator()( T a, T b ) { return ( max( a, b ) ); }
};
template<typename T = ll> class CompareMin {
public:	T operator()( T a, T b ) { return ( min( a, b ) ); }
};

template<typename Monoid, typename T = ll> struct SegTree {
	using Func = function<Monoid( Monoid, Monoid )>;
	const Func func;
	const Monoid UNITY;
	T SIZE_R;
	vector<Monoid> data;

	SegTree( T n, const Func f, const Monoid &unity ) :
		func( f ), UNITY( unity ) {
			init( n );
		}

	void init( T n ) {
		SIZE_R = 1;
		while ( SIZE_R < n ) SIZE_R *= 2;
		data.assign( SIZE_R * 2, UNITY );
	}

	void set( T idx, const Monoid &val ) { data[idx + SIZE_R] = val; }
	void build() {
		for ( T k = SIZE_R - 1; k > 0; --k )
			data[k] = func( data[k * 2], data[k * 2 + 1] );
	}

	void update( T idx, const Monoid &val ) {
		T k = idx + SIZE_R;
		data[k] = val;
		while ( k >>= 1 ) data[k] = func( data[k * 2], data[k * 2 + 1] );
	}

	Monoid get( T a, T b ) {
		Monoid vleft = UNITY, vright = UNITY;
		for ( T left = a + SIZE_R, right = b + SIZE_R;
			  left < right;
			  left >>= 1, right >>= 1 ) {
			if ( left & 1 ) vleft = func( vleft, data[left++] );
			if ( right & 1 ) vright = func( data[--right], vright );
		}

		return ( func( vleft, vright ) );
	}

	inline Monoid operator[]( ll idx ) { return ( data[idx + SIZE_R] ); }
};

template<typename T = ll> T solveLIS( const vector<T> &v ) {
	vector<T> dp( v.size(), numeric_limits<T>::max() );
	rep( i, v.size() ) {
		*lower_bound( ALL( dp ), v[i] ) = v[i];
	}
	return ( distance( dp.begin(), lower_bound( ALL( dp ), numeric_limits<T>::max() ) ) );
}

template<typename T, T MOD> struct Fp {
	T val;

	constexpr Fp( T input ) noexcept : val( input % MOD ) {
		if ( val < 0 ) val += MOD;
	}
	constexpr T getmod() { return ( MOD ); }
	constexpr Fp operator-() const noexcept {
		return ( val ? MOD - val : 0 );
	}
	constexpr Fp operator+( const Fp &r ) const noexcept { return ( ( Fp( *this ) += r ) ); }
	constexpr Fp operator-( const Fp &r ) const noexcept { return ( ( Fp( *this ) -= r ) ); }
	constexpr Fp operator*( const Fp &r ) const noexcept { return ( ( Fp( *this ) *= r ) ); }
	constexpr Fp operator/( const Fp &r ) const noexcept { return ( ( Fp( *this ) /= r ) ); }

	constexpr Fp &operator+=( const Fp &r ) noexcept {
		val += r.val;
		if ( val >= MOD ) val -= MOD;
		return ( *this );
	}

	constexpr Fp &operator-=( const Fp &r ) noexcept {
		val -= r.val;
		if ( val < 0 ) val += MOD;
		return ( *this );
	}

	constexpr Fp &operator*=( const Fp &r ) noexcept {
		val = ( val * r.val ) % MOD;
		return ( *this );
	}

	constexpr Fp &operator/=( const Fp &r ) noexcept {
		T a = r.val, b = MOD, u = 1, v = 0;
		while ( b ) {
			T t = a / b;
			a -= t * b; swap( a, b );
			u -= t * v; swap( u, v );
		}
		val = ( val * u ) % MOD;
		if ( val < 0 ) val += MOD;
		return ( *this );
	}

	constexpr bool operator==( const Fp &r ) noexcept {
		return ( val == r.val );
	}

	constexpr bool operator!=( const Fp &r ) noexcept {
		return ( val != r.val );
	}

	friend constexpr ostream& operator<<( ostream &os, const Fp<T, MOD>&x ) noexcept {
		return ( os << x.val );
	}

	friend constexpr Fp<T, MOD> modpow( const Fp<T, MOD> &a, T n ) noexcept {
		if ( n == 0 ) return ( 1 );
		auto t = modpow( a, n / 2 );
		t = t * t;
		if ( n & 1 ) t = t * a;
        return ( t );
	}
};

template<typename T, T MOD> struct BiCoef {
	using mint = Fp<T, MOD>;
	vector<mint> fact_, inv_, finv_;

	constexpr BiCoef() {}
	constexpr BiCoef( T n ) noexcept :
		fact_( n, 1 ), inv_( n, 1 ), finv_( n, 1 )
	{
		init( n );
	}

	constexpr void init( T n ) {
		fact_.assign( n, 1 ); inv_.assign( n, 1 ); finv_.assign( n, 1 );
		for ( T i = 2; i < n; i++ ) {
			fact_[i] = fact_[i - 1] * i;
			inv_[i] = -inv_[MOD % i] * ( MOD / i );
			finv_[i] = finv_[i - 1] * inv_[i];
       }
    }

    constexpr mint comb( T n, T k ) const noexcept {
		if ( n < k || n < 0 || k < 0 ) return ( 0 );
		if ( n - k < k ) k = n - k;
		return ( fact_[n] * finv_[k] * finv_[n - k] );
    }

    constexpr mint fact( T n ) const noexcept {
		if ( n < 0 ) return ( 0 );
		return ( fact_[n] ) ;
    }

    constexpr mint inv( T n ) const noexcept {
		if ( n < 0 ) return ( 0 );
		return ( inv_[n] );
    }

    constexpr mint finv( T n ) const noexcept {
		if ( n < 0 ) return ( 0 );
		return ( finv_[n] );
    }
};

constexpr ll MOD = 1000000007LL;
using mint = Fp<ll, MOD>;
BiCoef<ll, MOD> bc;

/*
	regex reg( R"(^(dream|dreamer|erase|eraser)+$)" );
	smatch m;

	if ( regex_match( s, m, reg ) )
	{
		cout << "YES" << endl;
	}
	else
	{
		cout << "NO" << endl;
	}
*/
/*
	cout << setprecision( 16 );
	cout << fixed << setprecision( 16 );
*/

template<typename T = int> vector<T> Zalgo( const string &S ) {
	T N = (T)S.size();
	vector<T> res(N);
	res[0] = N;
	T i = 1, j = 0;
	while ( i < N ) {
		while (i+j < N && S[j] == S[i+j]) ++j;
		res[i] = j;
		if ( j == 0 ) { ++i; continue; }
		int k = 1;
		while ( i + k < N && k + res[k] < j ) res[i + k] = res[k], ++k;
		i += k, j -= k;
	}
	return ( res );
}

void replace( string &s, string t, string r ) {
	string::size_type p = 0;
	while ( ( p = s.find( t, p ) ) != string::npos ) {
		s.replace( p, t.length(), r );
		p += r.length();
	}
}

template<typename T> auto binsearch_idx( vector<T> &v, T val ) {
	ll left = -1, right = v.size();
	while ( abs( right - left ) > 1 )
	{
		ll mid = ( left + right ) / 2;
		if ( val > v[mid] )
			left = mid;
		else
			right = mid;
	}

	return ( mp( v[right] == val, right ) );
}


int main()
{
	ll n, k; cin >> n >> k;
	vector<ll> a( n );
	rep( i, n ) cin >> a[i];
	vector<ll> sum( n + 1 );
	rep( i, n ) sum[i + 1] = sum[i] + a[i];

	map<ll, ll> m;
	queue<ll> q;
	ll ans = 0;
	rep( i, n + 1 )
	{
		ll tmp = ( sum[i] - i + k ) % k;
		ans += m[tmp]++;
		q.push( tmp );
		if ( q.size() >= k )
		{
			m[q.front()]--;
			q.pop();
		}
	}

	cout << ans << endl;

	return ( 0 );
}