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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
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
Sequence in context: A332715 A219107 A338498 * A004604 A246924 A274715
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 May 26 16:22 EDT 2024. Contains 372840 sequences. (Running on oeis4.)