Rabbit Constant

The limiting rabbit sequence written as a binary fraction 👁 0.1011010110110..._2
(OEIS A005614), where 👁 b_2
denotes a binary number (a number in base-2). The decimal value is

(OEIS A014565).

Amazingly, the rabbit constant is also given by the continued fraction [0; 👁 2^(F_0)
, 👁 2^(F_1)
, 👁 2^(F_2)
, 👁 2^(F_3)
, ...] = [2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184, ...] (OEIS A000301), where 👁 F_n
are Fibonacci numbers with 👁 F_0
taken as 0 (Gardner 1989, Schroeder 1991). Another amazing connection was discovered by S. Plouffe. Define the Beatty sequence 👁 {a_i}
by

where 👁 |_x_|
is the floor function and 👁 phi
is the golden ratio. The first few terms are 1, 3, 4, 6, 8, 9, 11, ... (OEIS A000201). Then

This is a special case of the Devil's staircase function with 👁 x=1/phi
.

The irrationality measure of 👁 R
is 👁 1+phi
(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 Resource. https://mathworld.wolfram.com/RabbitConstant.html

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