## Rabbit Constant

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

(Sloane's A014565).

Amazingly, the rabbit constant is also given by the Continued Fraction [0, , , , , ...], 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

where is the Floor Function and is the Golden Ratio. The first few terms are 1, 3, 4, 6, 8, 9, 11, ... (Sloane's A000201). Then

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.