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Evolution of finite sequences of integers
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In the March 1996 issue of the Gazette, [p. 36] the following conjecture by Sir Bryan Thwaites appeared: ‘Take any set of n rational numbers. Iterate.Abstract:
In the March 1996 issue of the Gazette, [p. 36] the following conjecture by Sir Bryan Thwaites appeared: ‘Take any set of n rational numbers. Form another set by taking the positive differences of successive members of the first set, the last such difference being formed from the last and first members of the original set. Iterate. Then in due course the set so formed will consist entirely of zeroes if and only if n is a power of two.’read more
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Unbounded Ducci Sequences
TL;DR: The Ducci sequence has been referred to as the N -number game as mentioned in this paper, where every integer string of length n will iterate to the zero-string in a finite number of steps if and only if n 1 Ω4 2 for some positive integer m. This result has been restated and reproved a number of times since then; see Ref.
A characterization for the length of cycles of the n - number ducci game
Neil J. Calkin,John G. Stevens +1 more
TL;DR: In this paper, the authors considered the Ducci game as a map over the finite field ZZ2 and characterized the period lengths through orders of minimal annihilating polynomials.
Asymptotic behavior of certain ducci sequences
TL;DR: In this paper, it is shown that for certain starting vectors of odd length, the corresponding sequence of iterates asymptotically approaches a periodic vector, but never becomes periodic itself.
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Promenade around Pascal triangle number motives
TL;DR: In this paper, the authors survey classical results and recent developments, old and new problems, conjectures and ideas selected from the endless theme of iterated application of the fundamental rule of addition.
References
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Book
Chaos and Fractals: New Frontiers of Science
TL;DR: This book discusses Fractal Image Compression, the Causality Principle, Deterministic Laws and Chaos, and the Backbone of Fractals.
Journal Article
Cellular automata machines
TL;DR: A cellular automata machine is a computer optimized for the simulation of cellular Automata that allows it to run thousands of times faster than a general-purpose computer of comparable cost programmed to do the same task.