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A221166 The infinite generalized Fibonacci word p^[2]. 6
0, 1, 0, 3, 0, 3, 2, 3, 0, 3, 0, 1, 0, 1, 2, 1, 0, 1, 0, 3, 0, 1, 0, 1, 2, 1, 2, 3, 2, 1, 2, 1, 0, 1, 0, 3, 0, 1, 0, 1, 2, 1, 0, 1, 0, 3, 0, 3, 2, 3, 0, 3, 0, 1, 0, 3, 0, 3, 2, 3, 2, 1, 2, 3, 2, 3, 0, 3, 0, 1, 0, 3, 0, 3, 2, 3, 0, 3, 0, 1, 0, 1, 2, 1, 0, 1, 0, 3, 0, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..89.

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.

EXAMPLE

The infinite Fibonacci word f^[2] is A003849. If we apply the morphism {1,0}->{0,2} we have 2, 0, 2, 2, 0 ,2 ... Prepending a 1 and replacing the sequence with the partial sums plus 1 (mod 4), applying operator sigma_1, we have 1, 3, 3, 1, 3, 3, 1, 1, 3, 1. Finally prepending 0 and replacing the that sequence with the partial sums (mod 4), applying operator sigma_0, we have the a(n). - R. J. Mathar, Jul 09 2013

MAPLE

# fibi and fibonni defined in A221150

fmorph := proc(n, i)

    if fibonni(n, i) = 0 then

        2;

    else

        0 ;

    end if;

end proc:

sigma1f := proc(n, i)

    if n = 0 then

        1;

    else

        1 + modp(add(fmorph(j, i), j=0..n-1), 4) ;

    end if;

end proc:

sigma01f := proc(n, i)

    if n = 0 then

        0;

    else

        modp(add(sigma1f(j, i), j=0..n-1), 4) ;

    end if;

end proc:

A221166 := proc(n)

    sigma01f(n, 2) ;

end proc: # R. J. Mathar, Jul 09 2013

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]]]]]];

fmorph[n_, i_] := If[fibonni[n, i] == 0, 2, 0];

sigma1f[n_, i_] := If[n == 0, 1, 1+Mod[Sum[fmorph[j, i], {j, 0, n-1}], 4]];

sigma01f[n_, i_] := If[n == 0, 0, Mod[Sum[sigma1f[j, i], {j, 0, n-1}], 4]];

a[n_] := sigma01f[n, 2]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 21 2017, after R. J. Mathar *)

CROSSREFS

Cf. A221166-A221171.

Sequence in context: A332715 A219107 A338498 * A004604 A246924 A274715

Adjacent sequences:  A221163 A221164 A221165 * A221167 A221168 A221169

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 04 2013

EXTENSIONS

Changed name from p^[1] to p^[2] because p^[1] could not be reproduced. - R. J. Mathar, Jul 09 2013

STATUS

approved

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Last modified June 17 00:17 EDT 2021. Contains 345080 sequences. (Running on oeis4.)