Problem:
Score : 800 points

Problem Statement

Find the number, modulo 998244353, of sequences of length N consisting of 0, 1 and 2 such that none of their contiguous subsequences totals to X.

Constraints

1 \leq N \leq 3000

1 \leq X \leq 2N

N and X are integers.

Input

Input is given from Standard Input in the following format:

N X

Output

Print the number, modulo 998244353, of sequences that satisfy the condition.

Sample Input 1

3 3

Sample Output 1

14

14 sequences satisfy the condition: (0,0,0),(0,0,1),(0,0,2),(0,1,0),(0,1,1),(0,2,0),(0,2,2),(1,0,0),(1,0,1),(1,1,0),(2,0,0),(2,0,2),(2,2,0) and (2,2,2).

Sample Input 2

8 6

Sample Output 2

1179

Sample Input 3

10 1

Sample Output 3

1024

Sample Input 4

9 13

Sample Output 4

18402

Sample Input 5

314 159

Sample Output 5

459765451

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;
using System.Collections.Generic;
using System.Diagnostics;
using Debug = System.Diagnostics.Trace;
using SB = System.Text.StringBuilder;
using static System.Math;
using Number = System.Int32;
using System.Numerics;
using static System.Numerics.BigInteger;
namespace Program {
	public class Solver {
		Random rnd = new Random(0);
		public void Solve() {
			var n = ri;
			var X = ri;
			ModInt ans = 0;
			var val = new long[n + 1];
			var binom = new BinomialCoefficient(n + 5);

			const long M = ModInt.Mod;
			//総和が X 未満
			{
				var dp = new long[X + 5];
				dp[0] = 1;
				for (int i = 0; i < n; i++)
					for (int j = X; j >= 0; j--) {
						dp[j + 2] += dp[j];
						if (dp[j + 2] >= M) dp[j + 2] -= M;
						dp[j + 1] += dp[j];
						if (dp[j + 1] >= M) dp[j + 1] -= M;
					}
				for (int j = 0; j < X; j++) ans += dp[j];

			}
			//総和が X 以上で 全部偶数かつ X が奇数
			if (X % 2 == 1) {
				for (int i = 0; i <= n; i++)
					if (i * 2 > X) val[i] += 1;
			}
			//3. 以下の 3 つを満たすやつ
			//  - 一番左の 1 から一番右の 1 までの parity が X と異なる
			//  - 一番左の 2 から一番右の 2 までが X-1 以下
			//  - 一番右の 2 から一番左の 1 までが X-1 以下
			{

				//dp[i][j] = i 個まで 2 を端に追加でき、合計 j 個
				var dp = Enumerate(n + 1, x => new long[n + 2]);
				if (X % 2 != 1) dp[(X - 2) / 2][1] += 1;
				for (int i = 2; i <= n; i++)
					for (int j = i; j <= X; j++)
						if (j % 2 != X % 2) dp[(X - 1 - j) / 2][i] = binom[i - 2, j - i].num;
				for (int i = 0; i <= n; i++) {
					for (int k = 0; k < 2; k++) {
						var ndp = new long[n + 2];
						for (int j = 0; j <= n; j++) {
							ndp[j] += dp[i][j];
							if (ndp[j] >= M) ndp[j] -= M;
							ndp[Min(n + 1, j + i + 1)] += M - dp[i][j];
							if (ndp[Min(n + 1, j + i + 1)] >= M) ndp[Min(n + 1, j + i + 1)] -= M;
						}
						for (int j = 0; j <= n; j++) {
							ndp[j + 1] += ndp[j];
							if (ndp[j + 1] >= M) ndp[j + 1] -= M;
						}
						dp[i] = ndp;
					}
					// Debug.WriteLine(dp[i].AsJoinedString());
					for (int j = 0; j <= n; j++) {
						val[j] += dp[i][j];
						if (val[j] >= M) val[j] -= M;
					}
				}
			}
			//一番左の 1 から一番右の 1 までの parity が X と異なる
			//総和が X 未満
			{
				var dp = new long[X + 5];
				var ep = new long[X + 5];
				var fp = new long[X + 5];
				dp[0] = 1;
				for (int i = 0; i < n; i++) {
					for (int j = X; j >= 0; j--) {
						dp[j + 2] += dp[j];
						if (dp[j + 2] >= M) dp[j + 2] -= M;
						ep[j + 1] += dp[j];
						if (ep[j + 1] >= M) ep[j + 1] -= M;
						ep[j + 2] += ep[j];
						if (ep[j + 2] >= M) ep[j + 2] -= M;
						ep[j + 1] += ep[j];
						if (ep[j + 1] >= M) ep[j + 1] -= M;
						if ((j + 1) % 2 != X % 2) {
							fp[j + 1] += ep[j];
							if (fp[j + 1] >= M) fp[j + 1] -= M;
							fp[j + 1] += dp[j];
							if (fp[j + 1] >= M) fp[j + 1] -= M;
						}
						fp[j + 2] += fp[j];
						if (fp[j + 2] >= M) fp[j + 2] -= M;
					}
				}
				for (int j = 0; j < X; j++) ans -= fp[j];
			}

			for (int i = 0; i <= n; i++) ans += binom[n, i] * val[i];
			Console.WriteLine(ans);
		}
		const long INF = 1L << 60; static int[] dx = { -1, 0, 1, 0 }; static int[] dy = { 0, 1, 0, -1 }; int ri { get { return sc.Integer(); } }
		long rl { get { return sc.Long(); } }
		double rd { get { return sc.Double(); } }
		string rs { get { return sc.Scan(); } }
		public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput()); static T[] Enumerate<T>(int n, Func<int, T> f) {
			var a = new T[n]; for (int i = 0; i < a.Length; ++i) a[i] = f(i); return a;
		}
		static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }
	}
}

#region main
static class Ex {
	static public string AsString(this IEnumerable<char> ie) { return new string(ie.ToArray()); }
	static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") {
		return string.Join(st, ie);
	}
	static public void Main() {
		Console.SetOut(new Program.IO.Printer(Console.OpenStandardOutput()) { AutoFlush = false });
		var solver = new Program.Solver();
		//* 
		var t = new System.Threading.Thread(solver.Solve, 200000000);
		t.Start();
		t.Join();
		//*/
		// solver.Solve();
		Console.Out.Flush();
	}
}
#endregion
#region Ex
namespace Program.IO {
	using System.IO;
	using System.Text;
	using System.Globalization;

	public class Printer : StreamWriter {
		public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
		public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
	}

	public class StreamScanner {
		public StreamScanner(Stream stream) { str = stream; }

		public readonly Stream str;
		private readonly byte[] buf = new byte[1024];
		private int len, ptr;
		public bool isEof = false;
		public bool IsEndOfStream { get { return isEof; } }

		private byte read() {
			if (isEof) return 0;
			if (ptr >= len) {
				ptr = 0;
				if ((len = str.Read(buf, 0, 1024)) <= 0) {
					isEof = true;
					return 0;
				}
			}
			return buf[ptr++];
		}

		public char Char() {
			byte b = 0;
			do b = read(); while ((b < 33 || 126 < b) && !isEof);
			return (char)b;
		}
		public string Scan() {
			var sb = new StringBuilder();
			for (var b = Char(); b >= 33 && b <= 126; b = (char)read()) sb.Append(b);
			return sb.ToString();
		}
		public string ScanLine() {
			var sb = new StringBuilder();
			for (var b = Char(); b != '\n' && b != 0; b = (char)read()) if (b != '\r') sb.Append(b);
			return sb.ToString();
		}
		public long Long() { return isEof ? long.MinValue : long.Parse(Scan()); }
		public int Integer() { return isEof ? int.MinValue : int.Parse(Scan()); }
		public double Double() { return isEof ? double.NaN : double.Parse(Scan(), CultureInfo.InvariantCulture); }
	}
}

#endregion

#region ModInt
/// <summary>
/// [0,<see cref="Mod"/>) までの値を取るような数
/// </summary>
public struct ModInt {
	/// <summary>
	/// 剰余を取る値．
	/// </summary>
	public const long Mod = 998244353;

	/// <summary>
	/// 実際の数値．
	/// </summary>
	public long num;
	/// <summary>
	/// 値が <paramref name="n"/> であるようなインスタンスを構築します．
	/// </summary>
	/// <param name="n">インスタンスが持つ値</param>
	/// <remarks>パフォーマンスの問題上，コンストラクタ内では剰余を取りません．そのため，<paramref name="n"/> ∈ [0,<see cref="Mod"/>) を満たすような <paramref name="n"/> を渡してください．このコンストラクタは O(1) で実行されます．</remarks>
	public ModInt(long n) { num = n; }
	/// <summary>
	/// このインスタンスの数値を文字列に変換します．
	/// </summary>
	/// <returns>[0,<see cref="Mod"/>) の範囲内の整数を 10 進表記したもの．</returns>
	public override string ToString() { return num.ToString(); }
	public static ModInt operator +(ModInt l, ModInt r) { l.num += r.num; if (l.num >= Mod) l.num -= Mod; return l; }
	public static ModInt operator -(ModInt l, ModInt r) { l.num -= r.num; if (l.num < 0) l.num += Mod; return l; }
	public static ModInt operator *(ModInt l, ModInt r) { return new ModInt(l.num * r.num % Mod); }
	public static implicit operator ModInt(long n) { n %= Mod; if (n < 0) n += Mod; return new ModInt(n); }

	/// <summary>
	/// 与えられた 2 つの数値からべき剰余を計算します．
	/// </summary>
	/// <param name="v">べき乗の底</param>
	/// <param name="k">べき指数</param>
	/// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
	public static ModInt Pow(ModInt v, long k) { return Pow(v.num, k); }

	/// <summary>
	/// 与えられた 2 つの数値からべき剰余を計算します．
	/// </summary>
	/// <param name="v">べき乗の底</param>
	/// <param name="k">べき指数</param>
	/// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
	public static ModInt Pow(long v, long k) {
		long ret = 1;
		for (k %= Mod - 1; k > 0; k >>= 1, v = v * v % Mod)
			if ((k & 1) == 1) ret = ret * v % Mod;
		return new ModInt(ret);
	}
	/// <summary>
	/// 与えられた数の逆元を計算します．
	/// </summary>
	/// <param name="v">逆元を取る対象となる数</param>
	/// <returns>逆元となるような値</returns>
	/// <remarks>法が素数であることを仮定して，フェルマーの小定理に従って逆元を O(log N) で計算します．</remarks>
	public static ModInt Inverse(ModInt v) { return Pow(v, Mod - 2); }
}
#endregion
#region Binomial Coefficient
public class BinomialCoefficient {
	public ModInt[] fact, ifact;
	public BinomialCoefficient(int n) {
		fact = new ModInt[n + 1];
		ifact = new ModInt[n + 1];
		fact[0] = 1;
		for (int i = 1; i <= n; i++)
			fact[i] = fact[i - 1] * i;
		ifact[n] = ModInt.Inverse(fact[n]);
		for (int i = n - 1; i >= 0; i--)
			ifact[i] = ifact[i + 1] * (i + 1);
		ifact[0] = ifact[1];
	}
	public ModInt this[int n, int r] {
		get {
			if (n < 0 || n >= fact.Length || r < 0 || r > n) return 0;
			return fact[n] * ifact[n - r] * ifact[r];
		}
	}
	public ModInt RepeatedCombination(int n, int k) {
		if (k == 0) return 1;
		return this[n + k - 1, k];
	}
}
#endregion