The limiting rabbit sequence written as a binary fraction (OEIS A005614 ),
where
denotes a binary number (a number in base-2). The decimal
value is
(1)
(OEIS A014565 ).
Amazingly, the rabbit constant is also given by the continued fraction [0; ,
, , , ...] = [2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184,
...] (OEIS A000301 ), where are Fibonacci numbers
with
taken as 0 (Gardner 1989, Schroeder 1991). Another amazing connection was discovered
by S. Plouffe. Define the Beatty sequence by
(2)
where
is the floor function and is the golden ratio . The
first few terms are 1, 3, 4, 6, 8, 9, 11, ... (OEIS A000201 ).
Then
(3)
This is a special case of the Devil's staircase function with .
The irrationality measure of is (D. Terr, pers. comm., May 21, 2004).
See also Devil's Staircase ,
Rabbit Sequence ,
Thue Constant ,
Thue-Morse
Constant
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References Anderson, P. G.; Brown, T. C.; and Shiue, P. J.-S. "A Simple Proof of a Remarkable Continued Fraction Identity." Proc.
Amer. Math. Soc. 123 , 2005-2009, 1995. Finch, S. R. "Prouhet-Thue-Morse
Constant." §6.8 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 436-441,
2003. Gardner, M. Penrose
Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed.
New York: W. H. Freeman, pp. 21-22, 1989. Schroeder, M. Fractals,
Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H.
Freeman, p. 55, 1991. Sloane, N. J. A. Sequences A000301 ,
A000201 /M2322, A005614 ,
and A014565 in "The On-Line Encyclopedia
of Integer Sequences." Referenced on Wolfram|Alpha Rabbit Constant
Cite this as:
Weisstein, Eric W. "Rabbit Constant."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/RabbitConstant.html
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