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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


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