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 A014565 Decimal expansion of rabbit constant. 17

%I

%S 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 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 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 Decimal expansion of rabbit constant.

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

%C 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

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

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

%H G. C. Greubel, <a href="/A014565/b014565.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 P. G. Anderson, T. C. Brown, 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 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p. 754.

%H 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 C. 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RabbitConstant.html">Rabbit Constant.</a>

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

%F Equals -1 + A073115.

%F From _Peter Bala_, Nov 04 2013: (Start)

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

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

%F 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 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 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 0.709803442861291314641787399444575597012502205767...

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

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

%o fa(y, N=17)=

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

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

%o L=0; R=1; yl=1; yr=y;

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

%o return( R )

%o }

%o a=0.5*fa(t) /* computation of 0.709803442861291314641... */

%o /* _Joerg Arndt_, Apr 15 2010 */

%Y Cf. A005614, A073115, A119809, A119812.

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Dec 11 1999

%E More terms from _Simon Plouffe_, Dec 11 1999

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