A074073 -id:A074073 - cp's OEIS frontend

A074072 Numerators of iterations of Thue-Morse sequence.

0, 1, 3, 105, 13515, 1771476585, 3804217000364127435, 140350834813144189858090274002849666665, 23879457134773177491341539034414728352082211931046871698790017205810876429515

Offset: 0

Eric W. Weisstein, Aug 18 2002

			Fractions begin with 0, 1/4, 3/8, 105/256, 13515/32768, 1771476585/4294967296, ...

Amiram Eldar, Table of n, a(n) for n = 0..12
Eric Weisstein's World of Mathematics, Thue-Morse Constant.

Cf. A074073 (denominators).

f[0] = 0; f[n_] := (2^(-2^n))*(2^(2^(n - 1)) - 1)*(2^(2^(n - 1))*f[n - 1] + 1); Numerator@ Array[f, 9, 0] (* Amiram Eldar, Oct 05 2023 *)

a(8) from Amiram Eldar, Oct 05 2023

A162634 Numerators of fractions with denominators A000215(n) approximating the Thue-Morse constant.

Original entry on oeis.org

1, 2, 7, 106, 27031, 1771476586, 7608434000728254871, 140350834813144189858090274002849666666, 47758914269546354982683078068829456704164423862093743397580034411621752859031

Offset: 0

Vladimir Shevelev, Jul 08 2009, Jul 14 2009

nonn

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.

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

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

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

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

Added more terms, Joerg Arndt, Mar 11 2013

A122497 Let f(S) denote the interchange of 1's and 2's in S. Let S_0 = 1, S_{N+1} = f(S_N).S_N, where the dot indicates concatenation. Sequence gives S_0.S_1.S_2.S_3....

Original entry on oeis.org

1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2

Offset: 1

Roger L. Bagula, Sep 15 2006

An alternating triangular Morse-Thue sequence based on A010060 using {1,2} instead of {0,1} substitutions.

			The first few S_i are:
1
2, 1
1, 2, 2, 1
2, 1, 1, 2, 1, 2, 2, 1
1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1

G. C. Greubel, Table of n, a(n) for the first 13 rows, flattened
Eric Weisstein's World of Mathematics, Thue-Morse Constant

Cf. A010060, A014571, A014572, A074072, A074073.

ThueMorse[n_, b_] := Nest[Flatten[ # /. {1 -> {1, 2}, 2 -> {2, 1}}] &, {b}, n] a = Table[ThueMorse[n, 1 + Mod[n, 2]], {n, 0, 7}] Flatten[a]

a(n) = A059448(n) + 1. - Filip Zaludek, Dec 10 2016

Edited by N. J. A. Sloane, May 22 2007

cp's OEIS Frontend

A074072 Numerators of iterations of Thue-Morse sequence.

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A162634 Numerators of fractions with denominators A000215(n) approximating the Thue-Morse constant.

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A122497 Let f(S) denote the interchange of 1's and 2's in S. Let S_0 = 1, S_{N+1} = f(S_N).S_N, where the dot indicates concatenation. Sequence gives S_0.S_1.S_2.S_3....

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