#1098982 (C#) No.3182 recurrence relation’s intersection sum

• 問題: No.3182 recurrence relation’s intersection sum
• ユーザー: tobisatis
• 提出日時: 2025-06-13 22:21:06
• 言語: C# (.NET 8.0.404)
• 結果: AC
• 実行時間: 537 ms / 2,000 ms
• コード長: 8,138 bytes
• コンパイル時間: 7,777 ms
• コンパイル使用メモリ: 169,944 KB
• 実行使用メモリ: 194,600 KB
• 最終ジャッジ日時: 2025-06-13 22:21:26
• 合計ジャッジ時間: 18,816 ms
• ジャッジサーバーID （参考情報）: judge3 / judge1

ファイルパターン	結果
sample	AC * 3
other	AC * 40

コンパイルメッセージ

  復元対象のプロジェクトを決定しています...
  /home/judge/data/code/main.csproj を復元しました (91 ミリ秒)。
  main -> /home/judge/data/code/bin/Release/net8.0/main.dll
  main -> /home/judge/data/code/bin/Release/net8.0/publish/

ソースコード

#nullable enable

using System.Numerics;

void Run()
{
    var k = Int();
    var l = long.Parse(String());
    var r = long.Parse(String());

    var a = new Matrix(new int[k + 4, k + 4]);
    for (var i = 0; i <= k; i++) for (var j = 0; j <= i; j++) a[i, j] = i.C(j);
    a[k + 1, k + 1] = k;
    a[k + 2, k] = 1;
    a[k + 2, k + 1] = 1;
    a[k + 2, k + 2] = k;
    a[k + 3, k + 2] = 1;
    a[k + 3, k + 3] = 1;
    var v0 = new Matrix(new int[k + 4, 1]);
    v0[0, 0] = 1;
    v0[k + 1, 0] = 1;
    v0[k + 2, 0] = 1;
    var vr = a.Power(r + 1) * v0;
    var vl = a.Power(l) * v0;
    ModInt ans = vr[k + 3, 0] - vl[k + 3, 0];
    Out(ans);
}

#region
var _io_ = new AtCoderIO(){ Backend = new() };
Run();
_io_.Backend.Flush();

string String() => _io_.Next();
int Int() => int.Parse(String());
void Out(object? x, string? sep = null) => _io_.Out(x, sep);

class AtCoderIO
{
    public required StandardIOBackend Backend { get; init; }

    ReadOnlyMemory<string> _input = Array.Empty<string>();
    int _iter = 0;
    public string Next()
    {
        while (_iter >= _input.Length) (_input, _iter) = (Backend.ReadLine().Split(' '), 0);
        return _input.Span[_iter++];
    }

    public void Out(object? x, string? separator = null)
    {
        if (x == null) return;
        separator ??= Environment.NewLine;
        if (x is System.Collections.IEnumerable a and not string)
        {
            var objects = a.Cast<object>();
            if (separator == Environment.NewLine && !objects.Any()) return;
            x = string.Join(separator, objects);
        }
        Backend.WriteLine(x);
    }
}

class StandardIOBackend
{
    readonly StreamReader _sr = new(Console.OpenStandardInput());
    readonly StreamWriter _sw = new(Console.OpenStandardOutput()) { AutoFlush = false };
    public string ReadLine() => _sr.ReadLine()!;
    public void WriteLine(object? value) => _sw.WriteLine(value);
    public void Flush() => _sw.Flush();
}
#endregion

static class Extensions
{
    public static T[] Repeat<T>(this int time, Func<T> F) => Enumerable.Range(0, time).Select(_ => F()).ToArray();
}

readonly record struct ModInt // :
    // IEqualityOperators<ModInt, ModInt, bool>,
    // IAdditiveGroup<ModInt>,
    // IMultiplicativeGroup<ModInt>
{
    public const int Mod = 998244353;
    int V { get; init; }
    public ModInt(long value)
    {
        var v = value % Mod;
        if (v < 0) v += Mod;
        V = (int)v;
    }
    static ModInt New(int value) => new(){ V = value };

    public static implicit operator ModInt(long v) => new(v);
    public static implicit operator int(ModInt modInt) => modInt.V;

    public static ModInt AdditiveIdentity => New(0);
    public static ModInt operator +(ModInt a, ModInt b)
    {
        var v = a.V + b.V;
        if (v >= Mod) v -= Mod;
        return New(v);
    }
    public ModInt AdditiveInverse()
    {
        if (V == 0) return AdditiveIdentity;
        return New(Mod - V);
    }
    public static ModInt operator -(ModInt a, ModInt b)
    {
        var v = a.V - b.V;
        if (v < 0) v += Mod;
        return New(v);
    }

    public static ModInt MultiplicativeIdentity => New(1);
    public static ModInt operator *(ModInt a, ModInt b) => New((int)((long)a.V * b.V % Mod));
    public ModInt MultiplicativeInverse() => V == 0 ? throw new DivideByZeroException() : Power(V, Mod - 2, Mod);
    public static ModInt operator /(ModInt a, ModInt b) => a * b.MultiplicativeInverse();

    static long Power(long v, ulong p, long mod)
    {
        var (res, k) = (1L, v);
        while (p > 0)
        {
            if ((p & 1) > 0) res = res * k % mod;
            k = k * k % mod;
            p >>= 1;
        }
        return res;
    }
    public ModInt Power(long p) => p < 0 ? MultiplicativeInverse().Power(-p) : Power(V, (ulong)p, Mod);

    public override string ToString() => V.ToString();
}

static class FactorialExtensions
{
    public static ModInt Factorial(this int value)
    {
        Extend(value);
        return value < 0 ? _inv.Span[-value] : _fac.Span[value];
    }
    public static ModInt P(this int n, int r)
    {
        if (r < 0 || r > n) return 0;
        if (n <= MaxN) return Factorial(n) * Factorial(r - n);
        ModInt res = 1;
        for (var i = 0; i < r; i++) res *= n - i;
        return res;
    }
    public static ModInt C(this int n, int r)
    {
        if (r < 0 || r > n) return 0;
        r = Math.Min(r, n - r);
        return P(n, r) * Factorial(-r);
    }
    public static ModInt H(this int n, int r) => C(r + n - 1, r);

    public static ModInt ModIntInverse(this int n)
    {
        if (n == 0) throw new DivideByZeroException();
        if (n < 0) return ModIntInverse(-n).AdditiveInverse();
        if (n > MaxN) return ((ModInt)n).MultiplicativeInverse();
        return Factorial(n - 1) * Factorial(-n);
    }

    // [x^k](1-x)^-n = nHk
    public static ModInt[] NegativeBinomialSeries(long n, int m)
    {
        var res = new ModInt[m + 1];
        res[0] = 1;
        for (var i = 1; i <= m; i++) res[i] = res[i - 1] * (n - 1 + i) * ModIntInverse(i);
        return res;
    }

    const int MaxN = (1 << 24) - 1;
    static ReadOnlyMemory<ModInt> _fac = new ModInt[]{ 1 };
    static ReadOnlyMemory<ModInt> _inv = new ModInt[]{ 1 };
    static void Extend(int q)
    {
        var l = _fac.Length;
        if (q < 0) q = -q;
        if (q < l || MaxN < q) return;
        while (l <= q) l <<= 1;
        var fac = new ModInt[l];
        var inv = new ModInt[l];
        var sf = fac.AsSpan();
        sf[0] = 1;
        for (var i = 1; i < sf.Length; i++) sf[i] = sf[i - 1] * i;
        var si = inv.AsSpan();
        si[l - 1] = sf[l - 1].Power(-1);
        for (var i = si.Length - 1; i > 0; i--) si[i - 1] = si[i] * i;
        (_fac, _inv) = (fac, inv);
    }
}

readonly struct Matrix : IAdditionOperators<Matrix, Matrix, Matrix>, IMultiplyOperators<Matrix, Matrix, Matrix>
{
    long[,] V { get; init; }
    public long this[int i, int j] { get => V[i, j]; set { V[i, j] = Mod(value); } }
    public int X => V.GetLength(0);
    public int Y => V.GetLength(1);

    const int p = 998244353;
    static long Mod(long v)
    {
        var res = v % p;
        if (res < 0) res += p;
        return res;
    }

    public Matrix(int[,] matrix)
    {
        var (x, y) = (matrix.GetLength(0), matrix.GetLength(1));
        var v = new long[x, y];
        for (var i = 0; i < x; i++) for (var j = 0; j < y; j++) v[i, j] = Mod(matrix[i, j]);
        V = v;
    }
    public static implicit operator Matrix(int[,] matrix) => new(matrix);
    static Matrix With(long[,] matrix) => new(){ V = matrix };

    public static Matrix IdentityMatrix(int size)
    {
        var v = new long[size, size];
        for (var i = 0; i < size; i++) v[i, i] = 1;
        return With(v);
    }

    public static Matrix operator +(Matrix a, Matrix b)
    {
        var (v, va, vb, x, y) = (new long[a.X, a.Y], a.V, b.V, a.X, a.Y);
        for (var i = 0; i < x; i++) for (var j = 0; j < y; j++)
        {
            var sum = va[i, j] + vb[i, j];
            if (sum > p) sum -= p;
            v[i, j] = sum;
        }
        return With(v);
    }

    public static Matrix operator *(Matrix a, Matrix b)
    {
        var (v, va, vb, x, y, z) = (new long[a.X, b.Y], a.V, b.V, a.X, b.Y, a.Y);
        for (var i = 0; i < x; i++) for (var k = 0; k < z; k++)
        {
            var c = va[i, k];
            if (c != 0) for (var j = 0; j < y; j++) v[i, j] += vb[k, j] * c % p;
        }
        for (var i = 0; i < x; i++) for (var j = 0; j < y; j++) v[i, j] = Mod(v[i, j]);
        return With(v);
    }

    public static Matrix operator *(Matrix a, long k)
    {
        var (v, va, x, y) = (new long[a.X, a.Y], a.V, a.X, a.Y);
        for (var i = 0; i < x; i++) for (var j = 0; j < y; j++) v[i, j] = va[i, j] * k % p;
        return With(v);
    }

    public Matrix Power(long k)
    {
        var c = IdentityMatrix(X);
        var c2 = this;
        while (k > 0)
        {
            if ((k & 1) > 0) c *= c2;
            c2 *= c2;
            k >>= 1;
        }
        return c;
    }
}