A Relationship Between Binomial Coefficients and Cubes of Lucas Numbers

For nonnegative integer l, L_l is the lth Lucas Number and \left( \beginarrayln\\i\endarray \right) = \fracn!i!\left( n - i \right)! is the binomial coeffient. For any nonnegative integer l, k and positive integer n, l(k, 3, n) denotes the convolution of sequence \left\ \left( \beginarrayln\\i\endarray \right) \right\_i = 0^n and \left\ L_k + i^3 \right\_i = 0^n, namely, l(k, 3, n)=\left( \beginarrayln\\0\endarray \right)L_k^3 + \left( \beginarrayln\\1\endarray \right)L_k + 1^3 + \cdots + \left( \beginarrayln\\n\endarray \right)L_k + n^3 = \sum\limits_i = 0^n \left( \beginarrayln\\i\endarray \right)L_k + i^3 . According to the definition of the Fibonacci sequence and by using the knowledge of elementary number theory, it is proved that l(k, 3, n) is equal to 2ⁿL_3k+2n+3(-1)^k+nL_k-n or 2ⁿL_3k+2n+3L_n-k when k≥n or not.