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%I #21 Nov 21 2017 03:08:52
%S 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,
%T 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,
%U 0,1,0,3,0,3,2,3,0,3,0,1,0,1,2,1,0,1,0,3,0,3
%N The infinite generalized Fibonacci word p^[2].
%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.
%e 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
%p # fibi and fibonni defined in A221150
%p fmorph := proc(n,i)
%p if fibonni(n,i) = 0 then
%p 2;
%p else
%p 0 ;
%p end if;
%p end proc:
%p sigma1f := proc(n,i)
%p if n = 0 then
%p 1;
%p else
%p 1 + modp(add(fmorph(j,i),j=0..n-1),4) ;
%p end if;
%p end proc:
%p sigma01f := proc(n,i)
%p if n = 0 then
%p 0;
%p else
%p modp(add(sigma1f(j,i),j=0..n-1),4) ;
%p end if;
%p end proc:
%p A221166 := proc(n)
%p sigma01f(n,2) ;
%p end proc: # _R. J. Mathar_, Jul 09 2013
%t 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]]];
%t 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]]]]]];
%t fmorph[n_, i_] := If[fibonni[n, i] == 0, 2, 0];
%t sigma1f[n_, i_] := If[n == 0, 1, 1+Mod[Sum[fmorph[j, i], {j, 0, n-1}], 4]];
%t sigma01f[n_, i_] := If[n == 0, 0, Mod[Sum[sigma1f[j, i], {j, 0, n-1}], 4]];
%t a[n_] := sigma01f[n, 2]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Nov 21 2017, after _R. J. Mathar_ *)
%Y Cf. A221166-A221171.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Jan 04 2013
%E 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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