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A014565 Decimal expansion of rabbit constant.

%I A014565 #114 Feb 16 2025 08:32:33
%S A014565 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,
%T A014565 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,
%U A014565 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
%N A014565 Decimal expansion of rabbit constant.
%C A014565 Davison shows that the continued fraction is (essentially) A000301 and proves that this constant is transcendental. - _Charles R Greathouse IV_, Jul 22 2013
%C A014565 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
%C A014565 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
%D A014565 Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 439.
%D A014565 M. Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise, New York: W. H. Freeman, 1991.
%H A014565 Sean A. Irvine and Joerg Arndt, <a href="/A014565/b014565.txt">Table of n, a(n) for n = 0..2000</a>
%H A014565 W. W. Adams and J. L. Davison, <a href="https://doi.org/10.1090/S0002-9939-1977-0441879-4">A remarkable class of continued fractions</a>, Proc. Amer. Math. Soc. 65 (1977), 194-198.
%H A014565 P. G. Anderson, T. C. Brown, and P. J.-S. Shiue, <a href="https://doi.org/10.1090/S0002-9939-1995-1249866-4">A simple proof of a remarkable continued fraction identity</a> Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
%H A014565 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p. 754.
%H A014565 J. L. Davison, <a href="https://doi.org/10.1090/S0002-9939-1977-0429778-5">A series and its associated continued fraction</a>, Proc. Amer. Math. Soc. 63 (1977), pp. 29-32.
%H A014565 Martin Griffiths, <a href="https://www.jstor.org/stable/23249532">96.12 The sum of a series: rational or irrational?</a>, The Mathematical Gazette, Vol. 96, No. 535 (2012), pp. 121-124.
%H A014565 Clark Kimberling and K. B. Stolarsky, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.3.267">Slow Beatty sequences, devious convergence, and partitional divergence</a>, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.
%H A014565 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RabbitConstant.html">Rabbit Constant.</a>
%H A014565 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A014565 Equals Sum_{n>=1} 1/2^b(n) where b(n) = floor(n*phi) = A000201(n).
%F A014565 Equals -1 + A073115.
%F A014565 From _Peter Bala_, Nov 04 2013: (Start)
%F A014565 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))).
%F A014565 r = Sum_{n >= 2} ( floor((n+1)*phi) - floor(n*phi) )/2^n = (1/2)*Sum_{n >= 1} A014675(n)/2^n.
%F A014565 r = Sum_{n >= 1} floor(n/phi)/2^n = Sum_{n >= 1} A005206(n-1)/2^n.
%F A014565 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.
%F A014565 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).
%F A014565 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)
%e A014565 0.709803442861291314641787399444575597012502205767...
%t A014565 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_ *)
%t A014565 RealDigits[ FromDigits[{Nest[Flatten[# /. {0 -> {1}, 1 -> {1, 0}}] &, {1}, 12], 0}, 2], 10, 111][[1]] (* _Robert G. Wilson v_, Mar 13 2014 *)
%t A014565 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 *)
%o A014565 (PARI) /* fast divisionless routine from fxtbook */
%o A014565 fa(y, N=17)=
%o A014565 { my(t, yl, yr, L, R, Lp, Rp);
%o A014565 /* as powerseries correct up to order fib(N+2)-1 */
%o A014565   L=0; R=1; yl=1; yr=y;
%o A014565   for(k=1, N, t=yr; yr*=yl; yl=t; Lp=R; Rp=R+yr*L; L=Lp; R=Rp; );
%o A014565   return( R )
%o A014565 }
%o A014565 a=0.5*fa(0.5) /* _Joerg Arndt_, Apr 15 2010 */
%o A014565 (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
%o A014565 (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
%Y A014565 Cf. A005614, A073115, A119809, A119812.
%K A014565 nonn,cons
%O A014565 0,1
%A A014565 _Eric W. Weisstein_, Dec 11 1999
%E A014565 More terms from _Simon Plouffe_, Dec 11 1999