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Author: MathProgrammer

Explanations/Explanations- 3/Fibonacci Sum Explanation.txt

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+
+F(n + 2) - 1 = F(n) + F(n - 1) + F(n - 2) + ... + F(1)
+
+This can be proven by induction.
+
+To find the n-th Fibonacci number, use matrix exponentiation.
+
+----------------------------------------------------------------------
+
+void load_identity(long long I[N][N])
+{
+    for(int i = 0; i < N; i++)
+        for(int j = 0; j < N; j++)
+            I[i][j] = (i == j ? 1 : 0);
+}
+
+void copy(long long A[N][N], long long B[N][N])
+{
+    for(int i = 0; i < N; i++)
+        for(int j = 0; j < N; j++)
+            B[i][j] = A[i][j];
+}
+
+void multiply(long long A[N][N], long long B[N][N])
+{
+    long long P[N][N];
+
+    for(int row = 0; row < N; row++)
+    {
+        for(int column = 0; column < N; column++)
+        {
+            P[row][column] = 0;
+
+            for(int i = 0; i < N; i++)
+                P[row][column] += (A[row][i]*B[i][column])%MOD;
+
+            P[row][column] %= MOD;
+        }
+    }
+
+    copy(P, A);
+}
+
+void power(long long X[N][N], long long power)
+{
+    long long result[N][N];
+    load_identity(result);
+
+    while(power)
+    {
+        if(power%2 == 1)
+            multiply(result, X);
+
+        multiply(X, X);
+        power = power >> 1;
+    }
+
+    copy(result, X);
+}
+
+long long fibo(long long n)
+{
+    long long fibonacci[N][N] = { {1, 0} };
+    long long matrix[N][N] = { {1, 1},
+                               {1, 0} };
+
+
+    if(n <= 1)
+        return n;
+
+    power(matrix, n - 1);
+    multiply(fibonacci, matrix);
+
+    return fibonacci[0][0];
+}
+
+void solve()
+{
+    long long n, m;
+    scanf("%lld %lld", &n, &m);
+
+    long long n_sum = fibo(n + 1) + MOD - 1;
+    long long m_sum = fibo(m + 2) + MOD - 1;
+
+    long long total = (m_sum + MOD - n_sum)%MOD;
+    printf("%lld\n", total);
+}
+
+int main()
+{
+    int no_of_test_cases;
+    scanf("%d", &no_of_test_cases);
+
+    while(no_of_test_cases--)
+        solve();
+
+    return 0;
+}