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Find the explicit form of the sequence b_n

Source: Austrian Mathematical Olympiad 2000, Part 2, D1, P1

June 28, 2011
algebrabinomial theoremalgebra proposed

Problem Statement

The sequence an is defined by a0=4,a1=1a_0 = 4, a_1 = 1 and the recurrence formula an+1=an+6an−1a_{n+1} = a_n + 6a_{n-1}. The sequence bnb_n is given by bn=∑k=0n(nk)ak.b_n=\sum_{k=0}^n \binom nk a_k. Find the coefficients α,β\alpha,\beta so that bnb_n satisfies the recurrence formula b_{n+1} = \alpha b_n + \beta b_{n-1}. Find the explicit form of bnb_n.
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