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Rabbit Sequence


Rabbit sequence recurrence plot

A sequence which arises in the hypothetical reproduction of a population of rabbits. Let the substitution system map 0->1 correspond to young rabbits growing old, and 1->10 correspond to old rabbits producing young rabbits. Starting with 0 and iterating using string rewriting gives the terms 1, 10, 101, 10110, 10110101, 1011010110110, .... A recurrence plot of the limiting value of this sequence is illustrated above.

Converted to decimal, this sequence gives 1, 2, 5, 22, 181, ... (OEIS A005203), with the nth term given by the recurrence relation

 a(n)=a(n-1)2^(F_(n-1))+a(n-2),

with a(0)=0, a(1)=1, and F_n the nth Fibonacci number.

The limiting sequence written as a binary fraction 0.1011010110110..._2 (OEIS A005614), where (a_n...a_1a_0)_2 denotes a binary number (i.e., a number written in base 2, so a_i=0 or 1), is called the rabbit constant.


See also

Fibonacci Number, Rabbit Constant, Thue-Morse Sequence

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References

Davison, J. L. "A Series and Its Associated Continued Fraction." Proc. Amer. Math. Soc. 63, 29-32, 1977.Gould, H. W.; Kim, J. B.; and Hoggatt, V. E. Jr. "Sequences Associated with t-ary Coding of Fibonacci's Rabbits." Fib. Quart. 15, 311-318, 1977.Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, p. 55, 1991.Sloane, N. J. A. Sequences A005203/M1539 and A005614 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Rabbit Sequence

Cite this as:

Weisstein, Eric W. "Rabbit Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RabbitSequence.html

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