An elementary proof that random Fibonacci sequences grow exponentially

Makover, Eran; McGowan, Jeffrey

doi:10.1016/j.jnt.2006.01.002

Cited by 13 publications

(8 citation statements)

References 5 publications

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“…This would imply that the variance of g n increases exponentially fast with growth rate at least twice the growth rate of the expected value. In [4] where the case p = 1/2 and λ = 1 is considered, the growth rate of the variance is proved to be equal to 1 + √ 5. It would be of interest to know better about the exact value of the variance for any p and, more generally, about the moments of higher order.…”

Section: Non-analyticity In the Neighbourhood Ofmentioning

confidence: 99%

Growth rate for the expected value of a generalized random Fibonacci sequence

Janvresse¹,

Rittaud²,

Rue³

2009

J. Phys. A: Math. Theor.

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A random Fibonacci sequence is defined by the relation g n = |g n−1 ± g n−2 |, where the ± sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p the probability of a +), and the recurrence relation is of the form g n = |λg n−1 ± g n−2 |. When λ ≥ 2 and 0 < p ≤ 1, we prove that the expected value of g n grows exponentially fast. When λ = λ k = 2 cos(π/k) for some fixed integer k ≥ 3, we show that the expected value of g n grows exponentially fast for p > (2 − λ k )/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in [5].

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Section: Non-analyticity In the Neighbourhood Ofmentioning

confidence: 99%

Growth rate for the expected value of a generalized random Fibonacci sequence

Janvresse¹,

Rittaud²,

Rue³

2009

J. Phys. A: Math. Theor.

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“…For the random Fibonacci sequence this result was reproved by Viswanath [3] and a simple proof was presented in the work of Makover and McGowan [4].…”

Section: Introductionmentioning

confidence: 95%

A Fibonacci type sequence with Prouhet-Thue-Morse coefficients

Lipka¹,

Ulas²

2021

Preprint

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Let t n = (−1) s 2 (n) , where s 2 (n) is the sum of binary digits function. The sequence (t n ) n∈N is the well-known Prouhet-Thue-Morse sequence. In this note we initiate the study of the sequence (h n ) n∈N , where h 0 = 0, h 1 = 1 and for n ≥ 2 we define h n recursively as follows:We prove several results concerning arithmetic properties of the sequence (h n ) n∈N . In particular, we prove non-vanishing of h n for n ≥ 5, automaticity of the sequence (h n (mod m)) n∈N for each m, and other results.

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“…was assessed under the assumption that + is chosen with probability p and − with 1 − p. Other efforts to understand such sequences can be found in [16]- [18].…”

Section: Random Sequences and Dynamical Systemsmentioning

confidence: 99%

Random fibonacci sequences and capacity/power scaling in cooperative multihop networks

Simmons

Coon

2017

2017 IEEE International Conference on Communications (ICC)

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In this paper, we analyze capacity and power scaling in multihop cooperative AF relay networks. An analytical framework for this task is developed by drawing a correspondence between random Fibonacci sequences and the end-to-end multihop system model. It turns out, the exponential growth rate of these interesting sequences can be employed to establish scaling laws, from which we conclude that it is possible to construct multihop cooperative AF networks that simultaneously avoid 1) exponential capacity decay and 2) exponential transmit power growth across the network. This is done by ensuring the network's Lyapunov exponent (a key observable studied in random dynamical system theory) is zero, which can be achieved by appropriately selecting the amplification factors at each of the relay nodes. Our results apply to both fixed-gain and variable-gain relaying. To conclude our work, we demonstrate the presented theory through numerical simulations.

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An elementary proof that random Fibonacci sequences grow exponentially

Cited by 13 publications

References 5 publications

Growth rate for the expected value of a generalized random Fibonacci sequence

Growth rate for the expected value of a generalized random Fibonacci sequence

A Fibonacci type sequence with Prouhet-Thue-Morse coefficients

Random fibonacci sequences and capacity/power scaling in cooperative multihop networks

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