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On one additive problem connected with expansions on linear recurrrent sequence
A. V. Shutov Khabarovsk Division of the Institute for Applied Mathematics,
Far Eastern Branch, Russian Academy of Sciences (Vladimir)
Abstract:
Let a1,…,ad be natural numbers satisfying condition a1≥a2≥…≥ad−1≥ad=1. Define sequence {Tn} using the linear recurrent relation Tn=a1Tn−1+a2Tn−2+…+adTn−d and initial conditions T0=1, Tn=1+a1Tn−1+a2Tn−2+…+anT0 for n<d. Let N(w) be a set of natural numbers for which the greedy expansion on the linear recurrent sequence {Tn} ends with some word w. Here w is chosen in such a way that so that the set N(w) is non-empty. We study the problem about the number rk(N) of representations of a natural number N in as the sum of k terms from N(w).
Using the previously obtained description of the sets N(w) in terms of shifts of tori of dimension d−1, an asymptotic formula for the number of representations rk(N) is obtained, and also found upper bounds for the number of representations.
Conditions on k that ensure the existence of considered representations for all sufficiently large natural numbers N are found. In particular, such representations exist if k≥1+(a1+1)m−d+1(a1+1)d−1a1, where m is the length of the fixed end w of the greedy expansion. In addition, an asymptotic formula is obtained for the average number of representations.
In conclusion, several unsolved problems are formulated.
Keywords:
linear recurrent sequences, greedy expansions, fixed last digits, linear additive problem.
Received: 25.04.2023 Accepted: 12.09.2023
Citation:
A. V. Shutov, “On one additive problem connected with expansions on linear recurrrent sequence”, Chebyshevskii Sb., 24:3 (2023), 228–241
Linking options:
https://www.mathnet.ru/eng/cheb1333 https://www.mathnet.ru/eng/cheb/v24/i3/p228
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