

A221152


The generalized Fibonacci word f^[5].


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
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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] = (iphi)/(i^2i1) (see Ramirez et al., page 8),
so the algebraic conjugate of alpha[i] is (2i1+sqrt(5))/(2i^22i2) 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. 347373, 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), 194198.
Cyril Allauzen, Une caractérisation simple des nombres de Sturm, Journal de Théorie des Nombres de Bordeaux 10, no 2 (1998), 237241.
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), 20052009.
T. C. Brown, A characterization of the quadratic irrationals, Canad. Math. Bull, 1991, 34(1), 3641.
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_{n1}S_{n2}; 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*n1))/( (10^A022095(3*n3)  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}] (* JeanFrançois Alcover, Nov 21 2017, after R. J. Mathar *)


CROSSREFS

Cf. A003849, A005614, A221150, A221151. A022095, A230900.
Sequence in context: A219463 A286688 A285125 * A302048 A093957 A283317
Adjacent sequences: A221149 A221150 A221151 * A221153 A221154 A221155


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Jan 03 2013


STATUS

approved



