cp's OEIS Frontend

A058842 From Renyi's "beta expansion of 1 in base 3/2": sequence gives a(1), a(2), ... where x(n) = a(n)/2^n, with 0 < a(n) < 2^n, a(1) = 1, a(n) = 3*a(n-1) modulo 2^n.

%I A058842 #18 Oct 08 2017 05:55:08
%S A058842 1,3,1,3,9,27,81,243,217,651,1953,1763,5289,15867,14833,44499,2425,
%T A058842 7275,21825,65475,196425,589275,1767825,5303475,15910425,47731275,
%U A058842 8976097,26928291,80784873,242354619,727063857,2181191571,6543574713
%N A058842 From Renyi's "beta expansion of 1 in base 3/2": sequence gives a(1), a(2), ... where x(n) = a(n)/2^n, with 0 < a(n) < 2^n, a(1) = 1, a(n) = 3*a(n-1) modulo 2^n.
%C A058842 Let r be a real number strictly between 1 and 2, x any real number between 0 and 1; define y = (y(i)) by x(0) = x; x(i+1) = r*x(i)-1 if r*x(i)>1 and r*x(i) otherwise; y(i) = integer part of x(i+1): y = (y(i)) is an infinite word on the alphabet (0,1). Here we take r = 3/2 and x = 1.
%C A058842 It seems that the sequence x(n) = a(n)/2^n which satisfies 0 < x(n) < 1 is not equidistributed in (0,1) and perhaps lim_{n -> infinity} Sum_{k=1..n} x(k)/n = C < 0.4 < 1/2. - _Benoit Cloitre_, Aug 27 2002
%D A058842 A. Renyi (1957), Representation for real numbers and their ergodic properties, Acta. Math. Acad. Sci. Hung., 8, 477-493.
%H A058842 Reinhard Zumkeller, <a href="/A058842/b058842.txt">Table of n, a(n) for n = 1..1000</a>
%F A058842 Let x(1)=1, x(n+1) = (3/2)*x(n) - floor((3/2)*x(n)); then a(n) = x(n)*2^n - _Benoit Cloitre_, Aug 27 2002
%t A058842 x[1] = 1; x[n_] := x[n] = (3/2)*x[n-1] - Floor[(3/2)*x[n-1]]; a[n_] := x[n+1]*2^(n); Table[a[n], {n, 1, 33}] (* _Jean-François Alcover_, Oct 13 2011, after _Benoit Cloitre_ *)
%o A058842 (Haskell)
%o A058842 import Data.Ratio ((%), numerator, denominator)
%o A058842 a058842 n = a058842_list !! (n-1)
%o A058842 a058842_list = map numerator (renyi 1 []) where
%o A058842    renyi :: Rational -> [Rational] -> [Rational]
%o A058842    renyi x xs = r : renyi r (x:xs) where
%o A058842       r = q - fromInteger ((numerator q) `div` (denominator q))
%o A058842       q = 3%2 * x
%o A058842 -- _Reinhard Zumkeller_, Jun 28 2011
%Y A058842 Cf. A058841, A058840.
%K A058842 nonn,nice,easy
%O A058842 1,2
%A A058842 Claude Lenormand (claude.lenormand(AT)free.fr), Jan 05 2001
%E A058842 More terms from Larry Reeves (larryr(AT)acm.org), Feb 22 2001