A221151 The generalized Fibonacci word f^[4].

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

Offset

0

Links

Table of n, a(n) for n=0..132.

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

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.

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], 2012-2014.

Formula

Set S_0=0, S_1=0001; 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 + 3)) - floor((n + 1)/(phi + 3)) 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 = 0.00010 00010 00100 00100 00100 .... then we have the series representation c = sum {n >= 1} 1/10^floor(n*(phi + 3)). An alternative representation is c = 9*sum {n >= 1} floor(n/(phi + 3)) /10^n.

The constant 9*c has the simple continued fraction representation [0; 1111, 10, 10^4, 10^5, 10^9, ..., 10^A000285(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^A000285(3*n-1))/( (10^A000285(3*n-3) - 1)*(10^A000285(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 10 million decimal places. Cf. A005614 and A221150. (End)

Maple

# fibi and fibonni implemented in A221150.
A221151 := proc(n)
    fibonni(n, 4) ;
end proc: # R. J. Mathar, Jul 09 2013

Mathematica

a[n_] := Floor[(n+2)/(GoldenRatio+3)] - Floor[(n+1)/(GoldenRatio+3)];
Table[a[n], {n, 0, 132}] (* Jean-François Alcover, Nov 16 2017 *)

Crossrefs

Cf. A003849, A005614, A221150, A000285, A005614, A221152, A230900.

Sequence in context: A101349 A295308 A284954 * A359474 A359429 A353470

Adjacent sequences: A221148 A221149 A221150 * A221152 A221153 A221154

Keyword

nonn

Author

N. J. A. Sloane, Jan 03 2013

Status

approved