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A058840
From Renyi's "beta expansion of 1 in base 3/2": sequence gives y(0), y(1), ...
4
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
0,1
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
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)
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
(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
CROSSREFS
Sequence in context: A329670 A183919 A355449 * A266155 A262683 A359816
KEYWORD
nonn,nice,easy
AUTHOR
Claude Lenormand (claude.lenormand(AT)free.fr), Jan 05 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Feb 22 2001
STATUS
approved