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A058840 From Renyi's "beta expansion of 1 in base 3/2": sequence gives y(0), y(1), ... 4

%I #23 Dec 07 2019 12:07:16

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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

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

%H Reinhard Zumkeller, <a href="/A058840/b058840.txt">Table of n, a(n) for n = 0..10000</a>

%H Aubrey J. Kempner, <a href="http://www.jstor.org/stable/2300532">Anormal Systems of Numeration</a>, American Mathematical Monthly, volume 43, number 10, December 1936, pages 610-617.

%t r = 3/2; x = 1; a[0] = a[1] = 1;

%t For[n = 2, n<105, n++, x = If[r x > 1, r x - 1, r x]; a[n] = Floor[r x]];

%t Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Dec 21 2018, a solution I owe to _Benoit Cloitre_ *)

%o (Haskell)

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

%o a058840 n = a058840_list !! n

%o a058840_list = 1 : renyi' 1 where

%o renyi' x = y : renyi' r where

%o (r, y) | q > 1 = (q - 1, 1)

%o | otherwise = (q, 0)

%o q = 3%2 * x

%o -- _Reinhard Zumkeller_, Jul 01 2011

%o (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

%Y Cf. A058841, A058842.

%K nonn,nice,easy

%O 0,1

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

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

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Last modified March 28 04:13 EDT 2024. Contains 371235 sequences. (Running on oeis4.)