mod = 998244353 NTT_friend = [120586241,167772161,469762049,754974721,880803841,924844033,943718401,998244353,1045430273,1051721729,1053818881] NTT_dict = {} for i in range(len(NTT_friend)): NTT_dict[NTT_friend[i]] = i NTT_info = [[20,74066978],[25,17],[26,30],[24,362],[23,211],[21,44009197],[22,663003469],[23,31],[20,363],[20,330],[20,2789]] def popcount(n): c=(n&0x5555555555555555)+((n>>1)&0x5555555555555555) c=(c&0x3333333333333333)+((c>>2)&0x3333333333333333) c=(c&0x0f0f0f0f0f0f0f0f)+((c>>4)&0x0f0f0f0f0f0f0f0f) c=(c&0x00ff00ff00ff00ff)+((c>>8)&0x00ff00ff00ff00ff) c=(c&0x0000ffff0000ffff)+((c>>16)&0x0000ffff0000ffff) c=(c&0x00000000ffffffff)+((c>>32)&0x00000000ffffffff) return c def topbit(n): h = n.bit_length() h -= 1 return h def prepared_fft(mod = 998244353): rank2 = NTT_info[NTT_dict[mod]][0] root,iroot = [0] * 30,[0] * 30 rate2,irate2= [0] * 30,[0] * 30 rate3,irate3= [0] * 30,[0] * 30 root[rank2] = NTT_info[NTT_dict[mod]][1] iroot[rank2] = pow(root[rank2],mod - 2,mod) for i in range(rank2-1,-1,-1): root[i] = root[i+1] * root[i+1] % mod iroot[i] = iroot[i+1] * iroot[i+1] % mod prod,iprod = 1,1 for i in range(rank2-1): rate2[i] = root[i + 2] * prod % mod irate2[i] = iroot[i + 2] * iprod % mod prod = prod * iroot[i + 2] % mod iprod = iprod * root[i + 2] % mod prod,iprod = 1,1 for i in range(rank2-2): rate3[i] = root[i + 3] * prod % mod irate3[i] = iroot[i + 3] * iprod % mod prod = prod * iroot[i + 3] % mod iprod = iprod * root[i + 3] % mod return root,iroot,rate2,irate2,rate3,irate3 root,iroot,rate2,irate2,rate3,irate3 = prepared_fft() def ntt(a): n = len(a) h = topbit(n) assert (n == 1 << h) le = 0 while le < h: if h - le == 1: p = 1 << (h - le - 1) rot = 1 for s in range(1 << le): offset = s << (h - le) for i in range(p): l = a[i + offset] r = a[i + offset + p] * rot % mod a[i + offset] = (l + r) % mod a[i + offset + p] = (l - r) % mod rot = rot * rate2[topbit(~s & -~s)] % mod le += 1 else: p = 1 << (h - le - 2) rot,imag = 1,root[2] for s in range(1 << le): rot2 = rot * rot % mod rot3 = rot2 * rot % mod offset = s << (h - le) for i in range(p): a0 = a[i + offset] a1 = a[i + offset + p] * rot a2 = a[i + offset + p * 2] * rot2 a3 = a[i + offset + p * 3] * rot3 a1na3imag = (a1 - a3) % mod * imag a[i + offset] = (a0 + a2 + a1 + a3) % mod a[i + offset + p] = (a0 + a2 - a1 - a3) % mod a[i + offset + p * 2] = (a0 - a2 + a1na3imag) % mod a[i + offset + p * 3] = (a0 - a2 - a1na3imag) % mod rot = rot * rate3[topbit(~s & -~s)] % mod le += 2 def intt(a): n = len(a) h = topbit(n) assert (n == 1 << h) coef = pow(n,mod - 2,mod) for i in range(n): a[i] = a[i] * coef % mod le = h while le: if le == 1: p = 1 << (h - le) irot = 1 for s in range(1 << (le - 1)): offset = s << (h - le + 1) for i in range(p): l = a[i + offset] r = a[i + offset + p] a[i + offset] = (l + r) % mod a[i + offset + p] = (l - r) * irot % mod irot = irot * irate2[topbit(~s & -~s)] % mod le -= 1 else: p = 1 << (h - le) irot,iimag = 1,iroot[2] for s in range(1 << (le - 2)): irot2 = irot * irot % mod irot3 = irot2 * irot % mod offset = s << (h - le + 2) for i in range(p): a0 = a[i + offset] a1 = a[i + offset + p] a2 = a[i + offset + p * 2] a3 = a[i + offset + p * 3] a2na3iimag = (a2 - a3) * iimag % mod a[i + offset] = (a0 + a1 + a2 + a3) % mod a[i + offset + p] = (a0 - a1 + a2na3iimag) * irot % mod a[i + offset + p * 2] = (a0 + a1 - a2 - a3) * irot2 % mod a[i + offset + p * 3] = (a0 - a1 - a2na3iimag) * irot3 % mod irot *= irate3[topbit(~s & -~s)] irot %= mod le -= 2 def convolute_naive(a,b): res = [0] * (len(a) + len(b) - 1) for i in range(len(a)): for j in range(len(b)): res[i+j] = (res[i+j] + a[i] * b[j] % mod) % mod return res def convolute(a,b): s = a[:] t = b[:] n = len(s) m = len(t) if min(n,m) <= 60: return convolute_naive(s,t) le = 1 while le < n + m - 1: le *= 2 s += [0] * (le - n) t += [0] * (le - m) ntt(s) ntt(t) for i in range(le): s[i] = s[i] * t[i] % mod intt(s) s = s[:n + m - 1] return s def fps_inv(f,deg = -1): assert (f[0] != 0) if deg == -1: deg = len(f) res = [0] * deg res[0] = pow(f[0],mod-2,mod) d = 1 while d < deg: a = [0] * (d << 1) tmp = min(len(f),d << 1) a[:tmp] = f[:tmp] b = [0] * (d << 1) b[:d] = res[:d] ntt(a) ntt(b) for i in range(d << 1): a[i] = a[i] * b[i] % mod intt(a) a[:d] = [0] * d ntt(a) for i in range(d << 1): a[i] = a[i] * b[i] % mod intt(a) for j in range(d,min(d << 1,deg)): if a[j]: res[j] = mod - a[j] else: res[j] = 0 d <<= 1 return res def fps_div(f,g): n,m = len(f),len(g) if n < m: return [],f rev_f = f[:] rev_f = rev_f[::-1] rev_g = g[:] rev_g = rev_g[::-1] rev_q = convolute(rev_f,fps_inv(rev_g,n-m+1))[:n-m+1] q = rev_q[:] q = q[::-1] p = convolute(g,q) r = f[:] for i in range(min(len(p),len(r))): r[i] -= p[i] r[i] %= mod while len(r): if r[-1] != 0: break r.pop() return q,r def Bostan_Mori(N,P,Q): # P(x) / Q(x) assert (len(P)) d = len(Q) - 1 n = N while True: if n == 0: return P[0] QQ = [Q[i] for i in range(d+1)] for i in range(1,d+1,2): QQ[i] = mod - QQ[i] UU = convolute(P,QQ) U_e = [] U_o = [] for i in range(len(UU)): if i % 2: U_o.append(UU[i]) else: U_e.append(UU[i]) V = convolute(Q,QQ) Q = [V[2*i] for i in range(d+1)] if n % 2: P = U_o[:] else: P = U_e[:] n //= 2 def gauss_jordan(A,b): n = len(A) for i in range(n): pivot = A[i][i] for j in range(i,n): A[i][j] = A[i][j] * pow(pivot,-1,mod) % mod b[i][0] = b[i][0] * pow(pivot,-1,mod) % mod for j in range(i + 1,n): pivot = A[j][i] for k in range(i,n): A[j][k] = (A[j][k] - pivot * A[i][k] % mod) % mod b[j][0] = (b[j][0] - b[i][0] * pivot % mod) % mod for i in range(n): for j in range(i): if A[j][i]: b[j][0] = (b[j][0] - A[j][i] * b[i][0] % mod) % mod A[j][i] = 0 return A,b def berlekamp_massey(A): n = len(A) B,C = [1],[1] l,m,p = 0,1,1 for i in range(n): d = A[i] for j in range(1,l + 1): d = (d + C[j] * A[i - j] % mod) % mod if d == 0: m += 1 continue T = C[:] q = pow(p,-1,mod) * d % mod while len(C) < len(B) + m: C.append(0) for j in range(len(B)): b = B[j] C[j + m] = (C[j + m] - q * b % mod) % mod if 2 * l <= i: B = T[:] l,m,p = i + 1 - l,1,d else: m += 1 return C N,K = map(int,input().split()) fib = [1,1] F = [1,1] for i in range(2,1000): fib.append((fib[-1] + fib[-2]) % mod) F.append(pow(fib[-1],K,mod)) g = berlekamp_massey(F) f = convolute(F,g)[:len(g) - 1] g = convolute(g,[1,mod - 1]) print(Bostan_Mori(N - 1,f,g))