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Let b(n) be a strictly increasing sequence of positive integers satisfyinglim_{n→∞} b(n+1)/b(n) = 1. Define a recurrence by taking a_n to be the element of b nearest to a_{n-1} + a_{n-2}, with ties broken by taking the smaller element. We prove that for any initial values a_1, a_2 in b, lim_{n→∞} a_n / a_{n-1} = φ := (...
Let b(n) be a strictly increasing sequence of positive integers satisfyinglim_{n→∞} b(n+1)/b(n) = 1. Define a recurrence by taking a_n to be the element of b nearest to a_{n-1} + a_{n-2}, with ties broken by taking the smaller element. We prove that for any initial values a_1, a_2 in b, lim_{n→∞} a_n / a_{n-1} = φ := (...
Let b(n) be a strictly increasing sequence of positive integers satisfyinglim_{n→∞} b(n+1)/b(n) = 1. Define a recurrence by taking a_n to be the element of b nearest to a_{n-1} + a_{n-2}, with ties broken by taking the smaller element. We prove that for any initial values a_1, a_2 in b, lim_{n→∞} a_n / a_{n-1} = φ := (...
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