The Prodiggers are quite a cool band and for this reason, they have been the surprise guest at the ENTER festival for the past 80 years. At the beginning of their careers, they weren’t so successful, so they had to spend time digging channels to earn money; hence the name. Anyway, they like to tour a lot and have surprising amounts of energy to do extremely long tours. However, they hate spending two consecutive days without having a concert, so they would like to avoid it.
A tour is defined by a sequence of concerts and days-off. You need to count in how many ways The Prodiggers can select k different tours of the same length between l and r.
For example if k = 2, l = 1 and r = 2, if we define concert day as {1} and day-off as {0}, here are all possible tours: {0}, {1}, {00}, {01}, {10}, {11}. But tour 00 can not be selected because it has 2 days-off in a row. Now, we need to count in how many ways we can select k = 2 tours of the same length in range [1;2]. Here they are: {0,1}; {01,10}; {01,11}; {10,11}.
Since their schedule is quite busy, they want you to tell them in how many ways can do that, modulo 1 000 000 007 (109 + 7).
Input
The first line of the input contains three integers k, l and r (1 ≤ k ≤ 200, 1 ≤ l ≤ r ≤ 1018).
Output
Output a single number: the number of ways to select k different tours of the same length, modulo 1 000 000 007.
Example
input
1 1 2
output
5
Solution:
import java.io.*;
import java.util.*;
public class A {
long mod = (long) 1e9 + 7;
class Num {
final long a, b;
public Num(long a, long b) {
this.a = a % mod;
this.b = b % mod;
}
public Num add(Num other) {
return new Num(a + other.a, b + other.b);
}
public Num multiply(Num other) {
return new Num(a * other.a + 5 * b * other.b, a * other.b + b * other.a);
}
}
int k;
long[][] coeffs;
void prepare() {
coeffs = new long[k + 1][k + 1];
long[][] tmpCoeffs = new long[k + 1][k + 1];
long factK = 1;
coeffs[0][0] = 1;
for (int free = 0; free < k; free++) {
factK = factK * (free + 1) % mod;
for (int i = 0; i <= k; i++) {
for (int j = 0; j <= k; j++) {
if (coeffs[i][j] != 0) {
tmpCoeffs[i + 1][j] += coeffs[i][j];
if (tmpCoeffs[i + 1][j] >= mod) {
tmpCoeffs[i + 1][j] -= mod;
}
tmpCoeffs[i][j + 1] += coeffs[i][j];
if (tmpCoeffs[i][j + 1] >= mod) {
tmpCoeffs[i][j + 1] -= mod;
}
long add = coeffs[i][j] * (mod - free) % mod;
tmpCoeffs[i][j] += add;
if (tmpCoeffs[i][j] >= mod) {
tmpCoeffs[i][j] -= mod;
}
}
}
}
for (int i = 0; i <= k; i++) {
for (int j = 0; j <= k; j++) {
coeffs[i][j] = tmpCoeffs[i][j];
tmpCoeffs[i][j] = 0;
}
}
}
factK = inv(factK);
for (int i = 0; i <= k; i++) {
for (int j = 0; j <= k; j++) {
coeffs[i][j] = coeffs[i][j] * factK % mod;
}
}
// System.err.println(Arrays.deepToString(coeffs));
pow1 = new Num[k + 1];
powA = new Num[k + 1];
pow2 = new Num[k + 1];
powB = new Num[k + 1];
pow1[0] = pow2[0] = powA[0] = powB[0] = new Num(1, 0);
for (int i = 1; i <= k; i++) {
pow1[i] = pow1[i - 1].multiply(PHI1);
pow2[i] = pow2[i - 1].multiply(PHI2);
powA[i] = powA[i - 1].multiply(A);
powB[i] = powB[i - 1].multiply(B);
}
}
long inv(long x) {
return pow(x, mod - 2);
}
long pow(long a, long b) {
long res = 1;
while (b != 0) {
if ((b & 1) == 1) {
res = res * a % mod;
}
a = a * a % mod;
b >>>= 1;
}
return res;
}
final Num PHI1 = new Num(inv(2), inv(2));
final Num PHI2 = new Num(inv(2), mod - inv(2));
final Num A = new Num(0, inv(5));
final Num B = new Num(0, mod - inv(5));
final Num ONE = new Num(1, 0);
Num[] pow1, pow2, powA, powB;
long get(long n) {
Num result = new Num(0, 0);
for (int i = 0; i <= k; i++) {
for (int j = 0; j <= k; j++) {
long coef = coeffs[i][j];
if (coef == 0) {
continue;
}
Num cur = new Num(coef, 0).multiply(powA[i]).multiply(powB[j]);
Num x = pow1[i].multiply(pow2[j]);
x = geomSum(x, n + 1);
result = result.add(cur.multiply(x));
}
}
return result.a;
}
Num geomSum(Num x, long n) {
if (n == 0) {
return new Num(0, 0);
}
if (n % 2 == 0) {
Num x2 = x.multiply(x);
Num result = geomSum(x2, n / 2);
return (ONE.add(x)).multiply(result);
} else {
return ONE.add(x.multiply(geomSum(x, n - 1)));
}
}
public void solve() {
k = in.nextInt();
prepare();
long l = in.nextLong(), r = in.nextLong();
// for (int i = 0; i < 20; i++) {
// Num fib = A.multiply(pow1[i]).add(B.multiply(pow2[i]));
// System.err.println(fib.a + " " + fib.b);
// System.err.println(get(i + 1) - get(i));
// }
out.println((get(r + 2) - get(l + 1) + mod) % mod);
}
public void run() {
in = new FastScanner();
out = new PrintWriter(System.out);
solve();
out.close();
}
FastScanner in;
PrintWriter out;
class FastScanner {
BufferedReader br;
StringTokenizer st;
public FastScanner(String fileName) {
try {
br = new BufferedReader(new FileReader(fileName));
} catch (FileNotFoundException e) {
}
}
public FastScanner() {
br = new BufferedReader(new InputStreamReader(System.in));
}
String nextToken() {
while (st == null || !st.hasMoreElements()) {
try {
st = new StringTokenizer(br.readLine());
} catch (IOException e) {
}
}
return st.nextToken();
}
int nextInt() {
return Integer.parseInt(nextToken());
}
long nextLong() {
return Long.parseLong(nextToken());
}
double nextDouble() {
return Double.parseDouble(nextToken());
}
}
public static void main(String[] args) {
new A().run();
}
}
