Lucas 1874 Fibonacci as binomial sum generalization problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg114760] Lucas 1874 Fibonacci as binomial sum generalization problem*From*: Roger Bagula <roger.bagula at gmail.com>*Date*: Thu, 16 Dec 2010 05:48:25 -0500 (EST)

Clear[t, n, m, k, a] t[n_, m_, k_] = Binomial[n - k*(m - 1), m - 1] a[n_, k_] = Sum[t[n, m, k], {m, 1, Floor[n/k]}] Table[Table[a[n, k], {n, 0, 20}], {k, 1, 21}] TableForm[%] The problem is getting a polynomial for the fifth row sequence. Rows by k and Characteristic polynomials found for them: k=1 x^2-x-1 k=2 x^3-x^2-1 k=3 x^4-x^3-1 k=4 x^3-x-1 k=5 ? {0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 6, 7, 8, 9, 10, 21, 27, 34, 42, 51, 71} Does anyone know a way to solve for the recursion or the polynomial associated with these row sequences? Roger Bagula