A221152 The generalized Fibonacci word f^[5].

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1

Offset

0

Comments

Shifted by 1, this is the binary sequence [(n+1)*alpha]-[n*alpha], n >= 1, where alpha = 1/(4+phi) has continued fraction [0,5,1,1,1,1,...]. Brown (1991, Theorem 3) shows that this is not fixed by any morphism 0 -> A, 1 -> B where A and B are finite binary strings. - N. J. A. Sloane, Sep 11 2016

In fact, none of the generalized Fibonacci words f^[i], and none of its shifts, are fixed by a morphism as soon as i>2. This follows from Allauzen's criterion for the f^[i]: they are Sturmian sequences with slope

alpha[i] = (i-phi)/(i^2-i-1) (see Ramirez et al., page 8),

so the algebraic conjugate of alpha[i] is (2i-1+sqrt(5))/(2i^2-2i-2) which lies in (0,1) for i>2. For the shifts of the f^[i] this follows from Yasutomi's work. - Michel Dekking, Apr 21 2018

References

S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, in Number theory and its applications (Kyoto, 1997), pp. 347-373, Kluwer Acad. Publ., Dordrecht, 1999.

Links

Table of n, a(n) for n=0..105.

W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.

Cyril Allauzen, Une caractérisation simple des nombres de Sturm, Journal de Théorie des Nombres de Bordeaux 10, no 2 (1998), 237-241.

P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity" Proc. Amer. Math. Soc. 123 (1995), 2005-2009.

T. C. Brown, A characterization of the quadratic irrationals, Canad. Math. Bull, 1991, 34(1), 36-41.

José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012.

Formula

Set S_0=0, S_1=00001; thereafter S_n = S_{n-1}S_{n-2}; sequence is S_{oo}.

From Peter Bala, Nov 19 2013: (Start)

a(n) = floor((n + 2)/(phi + 4)) - floor((n + 1)/(phi + 4)) where phi = 1/2*(1 + sqrt(5)) denotes the golden ratio.

If we read the present sequence as the digits of a decimal constant c = 0.00001 00000 10000 10000 01000 .... then we have the series representation c = sum {n >= 1} 1/10^floor(n*(phi + 4)). An alternative representation is c = 9*sum {n >= 1} floor(n/(phi + 4)) /10^n.

The constant 9*c has the simple continued fraction representation [0; 11111, 10, 10^5, 10^6, 10^11, ..., 10^A022095(n), ...] (see Adams and Davison).

Using this result we can find the alternating series representation c = 9*sum {n >= 1} (-1)^(n+1)*(1 + 10^A022095(3*n-1))/( (10^A022095(3*n-3) - 1)*(10^A022095(3*n) - 1) ).

The series converges very rapidly: for example, the first 10 terms of the series give a value for c accurate to more than 12 million decimal places. Cf. A005614, A221150 and A221151. (End)

Mathematica

fibi[n_, i_] := fibi[n, i] = Which[n == 0, {0}, n == 1, Append[Table[0, {j, 1, i - 1}], 1], True, Join[fibi[n - 1, i], fibi[n - 2, i]]];
fibonni[n_, i_] := fibonni[n, i] = Module[{fn, Fn}, For[fn = 0, True, fn++, Fn = fibi[fn, i]; If[ Length[ Fn] >= n + 1 && Length[Fn] > i + 3, Return[ Fn[[n + 1]]]]]];
a[n_] := fibonni[n, 5]; Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Nov 21 2017, after R. J. Mathar *)

Crossrefs

Cf. A003849, A005614, A221150, A221151. A022095, A230900.

Sequence in context: A356923 A356924 A285125 * A302048 A093957 A283317

Adjacent sequences: A221149 A221150 A221151 * A221153 A221154 A221155

Keyword

nonn

Author

N. J. A. Sloane, Jan 03 2013

Status

approved