Discovering linear-recurrent solutions to Hofstadter-like recurrences using symbolic computation

Fox, Nathan

doi:10.1016/j.jsc.2017.06.002

Cited by 4 publications

(5 citation statements)

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“…We are now able to describe an infinite family of solutions to the V -recurrence that consist of interleavings of five simpler sequences. These solutions are of a similar flavor to those in [7]. But, the methods of that paper would not find these solutions, as these include subsequences that are Θ( √ n) in growth.…”

Section: An Infinite Family Of Solutions To the V -Recurrencementioning

confidence: 94%

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On Some Solutions to Hofstadter's $V$-Recurrence

Alkan¹,

Aybar²,

Akdeniz³

2020

Preprint

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In this study, we explore the properties of certain solutions of Hofstadter's famous V -recurrence, defined by the nested recurrence relation. First, we discover the nature behind a finite chaotic meta-Fibonacci sequence in terms of mortality in the V -recurrence. Then, we construct a new kind of quasi-periodic solution which suggests a connection with another Hofstadter-Huber recursion, H(n) = H(n − H(n − 2)) + H(n − H(n − 3)).

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Section: An Infinite Family Of Solutions To the V -Recurrencementioning

confidence: 94%

“…Given a meta-Fibonacci recurrence, there is a known algorithm to search for solutions to it that satisfy a linear recurrence relation [7]. This algorithm finds infinite families of solutions that eventually consist of interleavings of simple (typically constant or linear) subsequences.…”

Section: A New Kind Of Solutionmentioning

confidence: 99%

On Some Solutions to Hofstadter's $V$-Recurrence

Alkan¹,

Aybar²,

Akdeniz³

2020

Preprint

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“…For example, 1, 1 is shorthand for Q * (1) = 1 and Q * (2) = 1, the initial condition for the Hofstadter Q-sequence. Sometimes, it is convenient to define Q * (n) = 0 for all n ≤ 0, as forcing sequences to die as previously described can limit the diversity of solutions we encounter [4,13]. This convention is noted with a symbol 0 followed by a semicolon at the start of the initial condition.…”

Section: Notationmentioning

confidence: 99%

“…Quasilinear sequences appear frequently in this paper. There are many known solutions to nested recurrences that are eventually quasilinear [4,9]. The quasilinear chunks of these solutions have a fixed starting point, but they continue forever.…”

Section: Future Workmentioning

confidence: 99%

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A New Approach to the Hofstadter $Q$-Recurrence

Fox

2018

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Nested recurrence relations are highly sensitive to their initial conditions. The bestknown nested recurrence, the Hofstadter Q-recurrence, generates sequences displaying a wide variety of behaviors. Most famous among these is the Hofstadter Q-sequence, which appears to be structured at a macro level and chaotic at a micro level. Other choices of initial conditions can lead to more predictable solutions, frequently interleavings of simple sequences. Previous work has focused on the form of a desired solution and on describing an initial condition that generates such a solution. In this paper, we flip this paradigm around. We illustrate how focusing on the form of an initial condition and describing the resulting sequences can yield strange families of new solutions to nested recurrences.

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