A058840 From Renyi's "beta expansion of 1 in base 3/2": sequence gives y(0), y(1), ...

1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 0

Author

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

Keywords

• nonn
• nice
• easy

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.

Kempner considers a "canonical" expansion of a real number in a non-integer base using the greedy algorithm. The greedy algorithm takes the largest possible integer digit in the range 0 <= digit < base at each digit position from high to low. For base 3/2, Kempner gives the present sequence of digits, except instead a(1)=0, as an example canonical 2 = 10.01000001001... Kempner notes too that a(1) omitted and the rest shifted down is a base-3/2 non-canonical 1 = .101000001001.... (canonical would be 1 = 1.000...). - Kevin Ryde, Dec 06 2019

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 = 0..10000
• Aubrey J. Kempner, Anormal Systems of Numeration, American Mathematical Monthly, volume 43, number 10, December 1936, pages 610-617.

Crossrefs

Cf. A058841, A058842.

Programs

Haskell

import Data.Ratio ((%), numerator, denominator)
a058840 n = a058840_list !! n
a058840_list = 1 : renyi' 1 where
   renyi' x = y : renyi' r  where
      (r, y) | q > 1     = (q - 1, 1)
             | otherwise = (q, 0)
      q = 3%2 * x
-- Reinhard Zumkeller, Jul 01 2011

Mathematica

r = 3/2; x = 1; a[0] = a[1] = 1;
For[n = 2, n<105, n++, x = If[r x > 1, r x - 1, r x]; a[n] = Floor[r x]];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 21 2018, a solution I owe to Benoit Cloitre *)

PARI

a_vector(len) = my(v=vector(len),c=2,d=1); for(i=1,len, if(c>=d,c-=d;v[i]=1); c*=3;d*=2); v; \\ Kevin Ryde, Dec 06 2019

Extensions

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