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A162634 Numerators of fractions with denominators A000215(n) approximating the Thue-Morse constant 1
1, 2, 7, 106, 27031, 1771476586, 7608434000728254871, 140350834813144189858090274002849666666, 47758914269546354982683078068829456704164423862093743397580034411621752859031 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

One can prove that if in the sequence of numbers N for which A010060(N+2^n)= A010060(N) you replace the odious (evil) terms by 1's (0's), then we obtain 2^(n+1)-periodic (0,1)-sequence; if you write it in the form .xx...,i.e., as a binary infinite fraction, then the corresponding fraction has the form a(n)/A000215(n). These fractions very fast converge to the Thue-Morse constant .4124540336401...; e.g a(5)/(2^32+1) approximates this constant up to 10^(-9). These approximations differ from A074072-A074073. Conjecture. For n>=1, the fraction a(n)/A000215(n) is a convergent corresponding to the continued fraction for the Thue-Morse constant.

LINKS

Table of n, a(n) for n=0..8.

V. Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence,

FORMULA

a(1)=2, and, for n>=2, a(n) = 1 + (2^(2^(n-1))-1) * a(n-1).

PROG

(PARI) a(n)=if(n<=1, [1, 2][n+1], 1+(2^(2^(n-1))-1)*a(n-1)); /* Joerg Arndt, Mar 11 2013 */

CROSSREFS

Cf. A010060, A000215, A085394, A085395, A081706, A161627, A161639, A074072, A074073.

Sequence in context: A122524 A229165 A307329 * A235470 A072664 A045310

Adjacent sequences:  A162631 A162632 A162633 * A162635 A162636 A162637

KEYWORD

nonn,uned

AUTHOR

Vladimir Shevelev, Jul 08 2009, Jul 14 2009

EXTENSIONS

Added more terms, Joerg Arndt, Mar 11 2013

STATUS

approved

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Last modified November 14 04:47 EST 2019. Contains 329108 sequences. (Running on oeis4.)