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A221166
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The infinite generalized Fibonacci word p^[2].
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6
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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
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OFFSET
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0,4
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LINKS
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EXAMPLE
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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
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MAPLE
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# 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:
sigma01f(n, 2) ;
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MATHEMATICA
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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]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Changed name from p^[1] to p^[2] because p^[1] could not be reproduced. - R. J. Mathar, Jul 09 2013
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STATUS
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approved
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