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Tuymaada 2022

Source: Tuymaada 2022 Junior P-8

September 12, 2022
number theorycombinatoricscounting

Problem Statement

Eight poles stand along the road. A sparrow starts at the first pole and once in a minute flies to a neighboring pole. Let a(n)a(n) be the number of ways to reach the last pole in 2n+12n + 1 flights (we assume a(m)=0a(m) = 0 for m<3m < 3). Prove that for all n≥4n \ge 4 a(n)−7a(n−1)+15a(n−2)−10a(n−3)+a(n−4)=0.a(n) - 7a(n-1)+ 15a(n-2) - 10a(n-3) +a(n-4)=0. (T. Amdeberhan, F. Petrov )
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