A058842 - OEIS

%I #18 Oct 08 2017 05:55:08

%S 1,3,1,3,9,27,81,243,217,651,1953,1763,5289,15867,14833,44499,2425,

%T 7275,21825,65475,196425,589275,1767825,5303475,15910425,47731275,

%U 8976097,26928291,80784873,242354619,727063857,2181191571,6543574713

%N 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 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 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 A. Renyi (1957), Representation for real numbers and their ergodic properties, Acta. Math. Acad. Sci. Hung., 8, 477-493.

%H Reinhard Zumkeller, <a href="/A058842/b058842.txt">Table of n, a(n) for n = 1..1000</a>

%F 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 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 (Haskell)

%o import Data.Ratio ((%), numerator, denominator)

%o a058842 n = a058842_list !! (n-1)

%o a058842_list = map numerator (renyi 1 []) where

%o    renyi :: Rational -> [Rational] -> [Rational]

%o    renyi x xs = r : renyi r (x:xs) where

%o       r = q - fromInteger ((numerator q) `div` (denominator q))

%o       q = 3%2 * x

%o -- _Reinhard Zumkeller_, Jun 28 2011

%Y Cf. A058841, A058840.

%K nonn,nice,easy

%O 1,2

%A Claude Lenormand (claude.lenormand(AT)free.fr), Jan 05 2001

%E More terms from Larry Reeves (larryr(AT)acm.org), Feb 22 2001