with .
The first few values are 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, ... (OEIS A005185;
Wolfram
2002, pp. 129-130, sequence (e); Wolfram 2022). These numbers
are /sometimes called -numbers. The Hofstadter -sequence can be implemented in the Wolfram
Language as
There are currently no rigorous analyses or detailed predictions of the rather erratic behavior of
(Guy 1994). It has, however, been demonstrated that the chaotic behavior of the -numbers shows some signs of order, namely
that they exhibit approximate period doubling,
self-similarity and scaling
(Pinn 1999, 2000). These properties are shared with the related sequence
with ,
which exhibits exact period doubling (Pinn 1999,
2000). The chaotic regions of are separated by predictable smooth behavior.
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. Monthly93, 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. Monthly98, 5-20, 1991.Pickover,
C. A. "The Crying of Fractal Batrachion ." Comput. & Graphics19, 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 ." Ch. 25 in Keys
to Infinity. New York: W. H. Freeman, pp. 183-191, 1995.Pinn,
K. "Order and Chaos is Hofstadter's Sequence." Complexity4, 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.
.
The first few values are 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, ... (OEIS A005185;
Wolfram
2002, pp. 129-130, sequence (e); Wolfram 2022). These numbers
are /sometimes called 
-numbers. The Hofstadter 
-sequence can be implemented in the Wolfram
Language as
(Guy 1994). It has, however, been demonstrated that the chaotic behavior of the 
-numbers shows some signs of order, namely
that they exhibit approximate period doubling,
self-similarity and scaling
(Pinn 1999, 2000). These properties are shared with the related sequence
,
which exhibits exact period doubling (Pinn 1999,
2000). The chaotic regions of 
are separated by predictable smooth behavior.
." Comput. & Graphics 19, 611-615,
1995. Reprinted in 
." Ch. 25 in 
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