Rabbit Constant

The limiting Rabbit Sequence written as a Binary Fraction $0.1011010110110\ldots_2$ (Sloane's A005614), where $b_2$ denotes a Binary number (a number in base-2). The Decimal value is

$$R=0.7098034428612913146\ldots$$

(Sloane's A014565).

Amazingly, the rabbit constant is also given by the Continued Fraction [0, $2^{F_0}$, $2^{F_1}$, $2^{F_2}$, $2^{F_3}$, ...], 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

$$a_i\equiv \left\lfloor{i\phi}\right\rfloor ,$$

where $\left\lfloor{x}\right\rfloor$ is the Floor Function and $\phi$ is the Golden Ratio. The first few terms are 1, 3, 4, 6, 8, 9, 11, ... (Sloane's A000201). Then

$$R=\sum_{i=1}^\infty 2^{-a_i}.$$

See also Rabbit Sequence, Thue Constant, Thue-Morse Constant

References

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html

Gardner, M. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. New York: W. H. Freeman, pp. 21-22, 1989.

Plouffe, S. ``The Rabbit Constant to 330 Digits.'' http://www.lacim.uqam.ca/piDATA/rabbit.txt.

Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 55, 1991.

Sloane, N. J. A. A005614, A014565, and A000201/M2322 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.