A014565 Decimal expansion of rabbit constant.

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

Offset: 0

Author

Eric W. Weisstein, Dec 11 1999

Keywords

• nonn
• cons

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

Examples

			0.709803442861291314641787399444575597012502205767...
		

References

• Steven 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, and 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.
• Martin Griffiths, 96.12 The sum of a series: rational or irrational?, The Mathematical Gazette, Vol. 96, No. 535 (2012), pp. 121-124.
• Clark 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.
• Index entries for transcendental numbers

Crossrefs

Cf. A005614, A073115, A119809, A119812.

Programs

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 *)

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.<A098317 (=> 298, 1259, 5331, ... digits). - M. F. Hasler, Nov 10 2018

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)

Extensions

More terms from Simon Plouffe, Dec 11 1999