A058842 - OEIS

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.

1, 3, 1, 3, 9, 27, 81, 243, 217, 651, 1953, 1763, 5289, 15867, 14833, 44499, 2425, 7275, 21825, 65475, 196425, 589275, 1767825, 5303475, 15910425, 47731275, 8976097, 26928291, 80784873, 242354619, 727063857, 2181191571, 6543574713

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OFFSET

1,2

COMMENTS

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.

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

REFERENCES

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

FORMULA

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

MATHEMATICA

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 *)

PROG

(Haskell)

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

a058842 n = a058842_list !! (n-1)

a058842_list = map numerator (renyi 1 []) where

renyi :: Rational -> [Rational] -> [Rational]

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

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

q = 3%2 * x

-- Reinhard Zumkeller, Jun 28 2011

CROSSREFS

Cf. A058841, A058840.