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BMT Algebra #4 - Sums with Golden Ratio

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October 11, 2020
Bmtalgebraratio

Problem Statement

Let φ\varphi be the positive solution to the equation x2=x+1.x^2=x+1. For n≥0n\ge 0, let ana_n be the unique integer such that φn−anφ\varphi^n-a_n\varphi is also an integer. Compute ∑n=010an.\sum_{n=0}^{10}a_n.
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