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%I #32 Dec 27 2020 19:30:54
%S 0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,1,0,
%T 1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,
%U 0,0,1,0,1,0,1,0,0,0,1,0,0,0,1,0,0
%N Trajectory of 0 under repeated application of the morphism 0 -> 0010, 1 -> 1010.
%C The morphism in the definition, 0 -> 0010, 1 -> 1010, is the square of the morphism tau: 0 -> 10, 1 -> 00.
%C This sequence is also the 0-limiting word of tau (see A284948).
%C It is also the image of A080426 under the morphism 1 -> 0,0,1,0; 3 -> 0,0,1,0,1,0,1,0.
%C This sequence underlies all of A297469, A298468, A328190, and A328196.
%C Theorem: a(n) = 1 iff the binary expansion of n-1 ends in an odd number of 0's (cf. A036554, A072939). For proof see comments by _Michel Dekking_ in A284948.
%C Is this A096268 with an additional 0 added in front? - _R. J. Mathar_, Nov 13 2019 [Yes: it follows, e.g., from the above theorem. - _Andrey Zabolotskiy_, Jan 12 2020]
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%p F(0):= (0,0,1,0): F(1):= (1,0,1,0):
%p B:= [0]: # if start at 0 get the present sequence, if start at 1 get A284948
%p for i from 1 to 4 do B:= map(F, B) od:
%p B;
%p # Or, construction via A080426:
%p f(1):= (1,3,1): f(3):= (1,3,3,3,1):
%p A:= [1]:
%p for i from 1 to 5 do A:= map(f, A) od:
%p A;
%p g(1):= (0,0,1,0); g(3):= (0,0,1,0,1,0,1,0);
%p map(g,A):
%t SubstitutionSystem[{0 -> {0, 0, 1, 0}, 1 -> {1, 0, 1, 0}}, 0, 4] // Last (* _Jean-François Alcover_, Apr 06 2020 *)
%Y Cf. A036554, A072939, A080426, A284948, A285384 (complement), A297469, A298468, A328190, A328196.
%K nonn
%O 1
%A _N. J. A. Sloane_, Nov 04 2019. Extensively revised Nov 05 2019 thanks to comments from _R. J. Mathar_.
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