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A058840
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From Renyi's "beta expansion of 1 in base 3/2": sequence gives y(0), y(1), ...
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4
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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
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0,1
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COMMENTS
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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
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REFERENCES
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A. Renyi (1957), Representation for real numbers and their ergodic properties, Acta. Math. Acad. Sci. Hung., 8, 477-493.
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LINKS
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MATHEMATICA
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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]];
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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Claude Lenormand (claude.lenormand(AT)free.fr), Jan 05 2001
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Feb 22 2001
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STATUS
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approved
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