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The generalized Fibonacci word f^[4].
6

%I #24 Nov 16 2017 07:17:10

%S 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,

%T 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,

%U 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

%N The generalized Fibonacci word f^[4].

%H W. W. Adams and J. L. Davison, <a href="http://www.jstor.org/stable/2041889">A remarkable class of continued fractions</a>, Proc. Amer. Math. Soc. 65 (1977), 194-198.

%H P. G. Anderson, T. C. Brown, P. J.-S. Shiue, <a href="http://people.math.sfu.ca/~vjungic/tbrown/tom-28.pdf">A simple proof of a remarkable continued fraction identity</a> Proc. Amer. Math. Soc. 123 (1995), 2005-2009.

%H José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, <a href="http://arxiv.org/abs/1212.1368">A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.

%F Set S_0=0, S_1=0001; thereafter S_n = S_{n-1}S_{n-2}; sequence is S_{oo}.

%F From _Peter Bala_, Nov 19 2013: (Start)

%F a(n) = floor((n + 2)/(phi + 3)) - floor((n + 1)/(phi + 3)) where phi = 1/2*(1 + sqrt(5)) denotes the golden ratio.

%F 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.

%F 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).

%F 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) ).

%F 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)

%p # fibi and fibonni implemented in A221150.

%p A221151 := proc(n)

%p fibonni(n,4) ;

%p end proc: # _R. J. Mathar_, Jul 09 2013

%t a[n_] := Floor[(n+2)/(GoldenRatio+3)] - Floor[(n+1)/(GoldenRatio+3)];

%t Table[a[n], {n, 0, 132}] (* _Jean-François Alcover_, Nov 16 2017 *)

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

%K nonn

%O 0

%A _N. J. A. Sloane_, Jan 03 2013