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
It remains unknown whether the recurrence defines for every positive integer (Deane and Gentile 2026). 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 ,
for which Pinn (2000) reports numerical evidence of generations of doubling length
and chaotic regions separated by 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.Deane, J. H. B. and Gentile, G. "Some Subsets
of Set
in the Diluted Hofstadter Problem." J. Diff. Eq. Appl.32, 223-252,
2026. https://doi.org/10.1080/10236198.2026.2617910.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 in Hofstadter's Sequence." Complexity4, 41-46, 1999.Pinn,
K. "A Chaotic Cousin of Conway's Recursive Sequence." Exp. 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.
-sequence
is the recursive sequence generated by the
recurrence equation
.
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
for every positive integer 
(Deane and Gentile 2026). 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
,
for which Pinn (2000) reports numerical evidence of generations of doubling length
and chaotic regions separated by smooth behavior.
in the Diluted Hofstadter Problem." J. Diff. Eq. Appl. 32, 223-252,
2026. 
." 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." Exp. Math. 9,
55-66, 2000.Sloane, N. J. A. Sequence