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Hofstadter 的 Q-序列

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HofstadterQ

由递推序列生成的递推方程

Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2)),

其中 Q(1)=Q(2)=1。前几个值是 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, ... (OEIS A005185; Wolfram 2002, pp. 129-130, 序列 (e); Wolfram 2022)。这些数字有时被称为 Q 数。Hofstadter Q 序列可以在 Wolfram 语言 中实现为

Hofstadter[1] = Hofstadter[2] = 1;
Hofstadter[n_Integer?Positive] := Hofstadter[n] = Block[
   {$RecursionLimit = Infinity},
   Hofstadter[n - Hofstadter[n - 1]] +
    Hofstadter[n - Hofstadter[n - 2]]
   ]

目前还没有对 Q(n) 的相当不规则行为进行严格的分析或详细的预测 (Guy 1994)。然而，已经证明，Q 数的混沌行为显示出一些有序的迹象，即它们表现出近似的周期倍增、自相似性和标度 (Pinn 1999, 2000)。这些性质与相关序列共享

D(n)=D(D(n-1))+D(n-1-D(n-2))

其中 D(1)=D(2)=1，它表现出精确的周期倍增 (Pinn 1999, 2000)。D(n) 的混沌区域被可预测的平滑行为分隔开。

另请参阅

Hofstadter-Conway $10,000 序列, Mallows 序列, 周期倍增

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参考文献

Conolly, B. W. "Fibonacci and Meta-Fibonacci Sequences." In Fibonacci and Lucas Numbers, and the Golden Section (Ed. S. Vajda). New York: Halstead Press, pp. 127-138, 1989.Dawson, R.; Gabor, G.; Nowakowski, R.; and Weins, D. "Random Fibonacci-Type Sequences." Fib. Quart. 23, 169-176, 1985.Guy, R. "Some Suspiciously Simple Sequences." Amer. Math. Monthly 93, 186-191, 1986.Guy, R. K. "Three Sequences of Hofstadter." §E31 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 231-232, 1994.Hofstadter, D. R. Gödel, Escher Bach: An Eternal Golden Braid. New York: Vintage Books, pp. 137-138, 1980.Kubo, T. and Vakil, R. "On Conway's Recursive Sequence." Disc. Math. 152, 225-252, 1996.Mallows, C. L. "Conway's Challenge Sequence." Amer. Math. Monthly 98, 5-20, 1991.Pickover, C. A. "The Crying of Fractal Batrachion 1489." Comput. & Graphics 19, 611-615, 1995. Reprinted in Chaos and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research (Ed. C. A. Pickover). Amsterdam, Netherlands: Elsevier, pp. 127-131, 1998.Pickover, C. A. "The Crying of Fractal Batrachion 1489." Ch. 25 in Keys to Infinity. New York: W. H. Freeman, pp. 183-191, 1995.Pinn, K. "Order and Chaos is Hofstadter's Q(n) Sequence." Complexity 4, 41-46, 1999.Pinn, K. "A Chaotic Cousin of Conway's Recursive Sequence." Exper. Math. 9, 55-66, 2000.Sloane, N. J. A. Sequence A005185/M0438 in "The On-Line Encyclopedia of Integer Sequences."Tanny, S. M. "A Well-Behaved Cousin of the Hofstadter Sequence." Disc. Math. 105, 227-239, 1992.Wolfram, S. "Recursive Sequences." A New Kind of Science. Champaign, IL: Wolfram Media, pp. 128-131, 2002.Wolfram, S. "What We've Learned from NKS Chapter 4: Systems Based on Numbers." Around minute 34:00. 2022. https://www.youtube.com/watch?v=2BbO5mr094A.

在 Wolfram|Alpha 上被引用

Hofstadter 的 Q-序列

请引用为

Weisstein, Eric W. "Hofstadter 的 Q-序列。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/HofstadtersQ-Sequence.html

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