The limiting rabbit sequence written as a binary fraction A005614 ),
where 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; A000301 ), where Fibonacci numbers
with Beatty sequence
(2)
where floor function and 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
See also Devil's Staircase ,
Rabbit Sequence ,
Thue Constant ,
Thue-Morse
Constant
Explore with Wolfram|Alpha
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. Gardner, M. Penrose
Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix, reissue ed. Schroeder, M. Fractals,
Chaos, Power Laws: Minutes from an Infinite Paradise. 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 https://mathworld.wolfram.com/RabbitConstant.html
Subject classifications More... Less...
(OEIS A005614),
where 
denotes a binary number (a number in base-2). The decimal
value is
,
, 
, 
, ...] = [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
is the floor function and 
is the golden ratio. The
first few terms are 1, 3, 4, 6, 8, 9, 11, ... (OEIS A000201).
Then
.
is 
(D. Terr, pers. comm., May 21, 2004).
