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• W2896622315 abstract "Given a recurrence sequence $H$, with $H_n = c_1 H_{n-1} + dots + c_t H_{n-t}$ where $c_i in mathbb{N}_0$ for all $i$ and $c_1, c_t geq 1$, the generalized Zeckendorf decomposition (gzd) of $m in mathbb{N}_0$ is the unique representation of $m$ using $H$ composed of blocks lexicographically less than $sigma = (c_1, dots, c_t)$. We prove that the gzd of $m$ uses the fewest number of summands among all representations of $m$ using $H$, for all $m$, if and only if $sigma$ is weakly decreasing. We develop an algorithm for moving from any representation of $m$ to the gzd, the analysis of which proves that $sigma$ weakly decreasing implies summand minimality. We prove that the gzds of numbers of the form $v_0 H_n + dots + v_ell H_{n-ell}$ converge in a suitable sense as $n to infty$, furthermore we classify three distinct behaviors for this convergence. We use this result, together with the irreducibility of certain families of polynomials, to exhibit a representation with fewer summands than the gzd if $sigma$ is not weakly decreasing." @default.
• W2896622315 created "2018-10-26" @default.
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• W2896622315 date "2018-10-09" @default.
• W2896622315 modified "2023-10-14" @default.
• W2896622315 title "Summand minimality and asymptotic convergence of generalized Zeckendorf decompositions" @default.
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• W2896622315 doi "https://doi.org/10.1007/s40993-018-0137-7" @default.
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