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 {a_i}" src="https://mathworld.wolfram.com/images/equations/RabbitConstant/Inline9.gif" style="height:15px; width:21px" /> 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).

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."