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A014565 Decimal expansion of rabbit constant. 18
7, 0, 9, 8, 0, 3, 4, 4, 2, 8, 6, 1, 2, 9, 1, 3, 1, 4, 6, 4, 1, 7, 8, 7, 3, 9, 9, 4, 4, 4, 5, 7, 5, 5, 9, 7, 0, 1, 2, 5, 0, 2, 2, 0, 5, 7, 6, 7, 8, 6, 0, 5, 1, 6, 9, 5, 7, 0, 0, 2, 6, 4, 4, 6, 5, 1, 2, 8, 7, 1, 2, 8, 1, 4, 8, 4, 6, 5, 9, 6, 2, 4, 7, 8, 3, 1, 6, 1, 3, 2, 4, 5, 9, 9, 9, 3, 8, 8, 3, 9, 2, 6, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Davison shows that the continued fraction is (essentially) A000301 and proves that this constant is transcendental. - Charles R Greathouse IV, Jul 22 2013

Using Davison's result we can find an alternating series representation for the rabbit constant r as r = 1 - sum {n >= 1} (-1)^(n+1)*(1 + 2^Fibonacci(3*n+1))/( (2^(Fibonacci(3*n - 1)) - 1)*(2^(Fibonacci(3*n + 2)) - 1) ). The series converges rapidly: for example, the first 10 terms of the series give a value for r accurate to more than 1.7 million decimal places. See A005614. - Peter Bala, Nov 11 2013

The rabbit constant is the number having the infinite Fibonacci word A005614 as binary expansion; its continued fraction expansion is A000301 = 2^A000045 (after a leading zero, depending on convention). - M. F. Hasler, Nov 10 2018

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 439.

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

LINKS

Sean A. Irvine and Joerg Arndt, Table of n, a(n) for n = 0..2000

W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.

P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity Proc. Amer. Math. Soc. 123 (1995), 2005-2009.

Joerg Arndt, Matters Computational (The Fxtbook), p. 754.

J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc. 63 (1977), pp. 29-32.

C. Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.

Eric Weisstein's World of Mathematics, Rabbit Constant.

FORMULA

Equals Sum_{n>=1} 1/2^b(n) where b(n) = floor(n*phi) = A000201(n).

Equals -1 + A073115.

From Peter Bala, Nov 04 2013: (Start)

The results of Adams and Davison 1977 can be used to find a variety of alternative series representations for the rabbit constant r. Here are several examples (phi denotes the golden ratio (1/2)*(1 + sqrt(5))).

r = Sum_{n >= 2} ( floor((n+1)*phi) - floor(n*phi) )/2^n = (1/2)*Sum_{n >= 1} A014675(n)/2^n.

r = Sum_{n >= 1} floor(n/phi)/2^n = Sum_{n >= 1} A005206(n-1)/2^n.

r = ( Sum_{n >= 1} 1/2^floor(n/phi) ) - 2 and r = ( Sum_{n >= 1} floor(n*phi)/2^n ) - 2 = ( Sum_{n >= 1} A000201(n)/2^n ) - 2.

More generally, for integer N >= -1, r = ( Sum_{n >= 1} 1/2^floor(n/(phi + N)) ) - (2*N + 2) and for all integer N, r = ( Sum_{n >= 1} floor(n*(phi + N))/2^n ) - (2*N + 2).

Also r = 1 - Sum_{n >= 1} 1/2^floor(n*phi^2) = 1 - Sum_{n >= 1} 1/2^A001950(n) and r = 1 - Sum_{n >= 1} floor(n*(2 - phi))/2^n = 1 - Sum_{n >= 1} A060144(n)/2^n. (End)

EXAMPLE

0.709803442861291314641787399444575597012502205767...

MATHEMATICA

Take[ RealDigits[ Sum[N[1/2^Floor[k*GoldenRatio], 120], {k, 0, 300}]-1][[1]], 103] (* Jean-François Alcover, Jul 28 2011, after Benoit Cloitre *)

RealDigits[ FromDigits[{Nest[Flatten[# /. {0 -> {1}, 1 -> {1, 0}}] &, {1}, 12], 0}, 2], 10, 111][[1]] (* Robert G. Wilson v, Mar 13 2014 *)

digits = 103; dm = 10; Clear[xi]; xi[b_, m_] := xi[b, m] = RealDigits[ ContinuedFractionK[1, b^Fibonacci[k], {k, 0, m}], 10, digits] // First; xi[2, dm]; xi[2, m = 2 dm]; While[xi[2, m] != xi[2, m - dm], m = m + dm]; xi[2, m] (* Jean-François Alcover, Mar 04 2015, update for versions 7 and up, after advice from Oleg Marichev *)

PROG

(PARI) /* fast divisionless routine from fxtbook */

fa(y, N=17)=

{ my(t, yl, yr, L, R, Lp, Rp);

/* as powerseries correct up to order fib(N+2)-1 */

  L=0; R=1; yl=1; yr=y;

  for(k=1, N, t=yr; yr*=yl; yl=t; Lp=R; Rp=R+yr*L; L=Lp; R=Rp; );

  return( R )

}

a=0.5*fa(0.5) /* Joerg Arndt, Apr 15 2010 */

(PARI) my(r=1, p=(3-sqrt(5))/2, n=1); while(r>r-=1.>>(n\p), n++); A014565=r \\ M. F. Hasler, Nov 10 2018

(PARI) my(f(n)=1.<<fibonacci(n)-1, g(n)=(f(n+2)+2)/f(n)/f(n+3)); 1-g(2)+g(5)-g(8) \\ Illustration of formula from Bala's comment. Using g(8) gives 70 digits; subsequent terms (+g(11), -g(14), +g(17), ...) each multiply the precision by 4.236 ~ A098317 (=> 298, 1259, 5331, ... digits). - M. F. Hasler, Nov 10 2018

CROSSREFS

Cf. A005614, A073115, A119809, A119812.

Sequence in context: A021589 A093444 A021858 * A073115 A176444 A197025

Adjacent sequences:  A014562 A014563 A014564 * A014566 A014567 A014568

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Dec 11 1999

EXTENSIONS

More terms from Simon Plouffe, Dec 11 1999

STATUS

approved

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Last modified December 13 22:42 EST 2018. Contains 318087 sequences. (Running on oeis4.)