A058840 - OEIS

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

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OFFSET

0,1

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.

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 *)

PROG

(Haskell)

import data.ratio ((%), numerator, denominator)