// yukicoder: No.621 3 x N グリッド上のドミノの置き方の数 // bal4u 2019.8.25 // OEIS A288028: Number of maximal matchings in the grid graph P_3 X P_n. // a(n) = a(n-1)+5*a(n-2)+11*a(n-3)+5*a(n-4)+14*a(n-5)+8*a(n-6)+3*a(n-7) // -5*a(n-9)-11*a(n-10)-a(n-11)+2*a(n-12)+a(n-15) for n>15. #include #include typedef long long ll; #define MOD 1000000007 //// 行列のべき乗 (1-indexedに注意) #define SZ 17 // 行列の大きさ指定(メモリ確保) #define MSZ (sizeof(com)) // 行列の大きさ int com[SZ][SZ]; // (N x N)行列 と (N x 1)行列の積。a, bは書き換えられない void mat_mul1(int ab[SZ][SZ], int a[SZ][SZ], int b[SZ][SZ], int n) { int i, r; memset(ab, 0, MSZ); for (r = 1; r <= n; r++) for (i = 1; i <= n; i++) ab[r][1] = (ab[r][1] + (ll)a[r][i] * b[i][1]) % MOD; } // (N x N)行列 と (N x N)行列の積。a, bは書き換えられない void mat_mul(int ab[SZ][SZ], int a[SZ][SZ], int b[SZ][SZ], int n) { int i, r, c; memset(ab, 0, MSZ); for (r = 1; r <= n; r++) for (i = 1; i <= n; i++) for (c = 1; c <= n; c++) ab[r][c] = (ab[r][c] + (ll)a[r][i] * b[i][c]) % MOD; } // (N x N)行列のべき乗p。aは書き換えられる! void mat_pow(int ap[SZ][SZ], int a[SZ][SZ], int n, ll p) { int i, tmp[SZ][SZ]; printf("mat_pow p=%lld\n", p); memset(ap, 0, MSZ); for (i = 1; i <= n; i++) ap[i][i] = 1; while (p > 0) { if (p & 1) mat_mul(tmp, ap, a, n), memcpy(ap, tmp, MSZ); mat_mul(tmp, a, a, n), memcpy(a, tmp, MSZ); p >>= 1; } } //// 本問題関連 int a[SZ][SZ]; int tbl[16] = {0,2,5,22,75,264,941,3286,11623,40960, /* 10 */ 144267,508812,1792981,6319994,22277291,78518760}; /* * a(n) = a(n-1)+5*a(n-2)+11*a(n-3)+5*a(n-4)+14*a(n-5)+8*a(n-6)+3*a(n-7) * -5*a(n-9)-11*a(n-10)-a(n-11)+2*a(n-12)+a(n-15) for n>15. */ int a[SZ][SZ] = {{0}, { 0, 1, 5,11, 5,14, 8, 3, 0,-5,-11,-1,2, 0, 0, 1 }, { 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 }, { 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 }, { 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 }, { 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 }, { 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 }, { 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 }, { 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 }, { 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 }, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 }, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 }, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 }, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 }, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 }, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 }}; int main() { ll N; int i, ans; int aa[SZ][SZ], b[SZ][SZ], c[SZ][SZ]; scanf("%lld", &N); if (N <= 15) ans = tbl[N]; else { mat_pow(aa, a, 15, N-15); // 行列のべき乗 memset(b, 0, MSZ); for (i = 1; i <= 15; i++) b[16-i][1] = tbl[i]; mat_mul1(c, aa, b, 15); ans = c[1][1]; if (ans < 0) ans += MOD; } printf("%d\n", ans); return 0; }