A328979 Trajectory of 0 under repeated application of the morphism 0 -> 0010, 1 -> 1010.

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, 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, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0

Offset

1

Comments

The morphism in the definition, 0 -> 0010, 1 -> 1010, is the square of the morphism tau: 0 -> 10, 1 -> 00.

This sequence is also the 0-limiting word of tau (see A284948).

It is also the image of A080426 under the morphism 1 -> 0,0,1,0; 3 -> 0,0,1,0,1,0,1,0.

This sequence underlies all of A297469, A298468, A328190, and A328196.

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.

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]

Links

Table of n, a(n) for n=1..85.

Index entries for sequences that are fixed points of mappings

Maple

F(0):= (0, 0, 1, 0): F(1):= (1, 0, 1, 0):
B:= [0]: # if start at 0 get the present sequence, if start at 1 get A284948
for i from 1 to 4 do B:= map(F, B) od:
B;
# Or, construction via A080426:
f(1):= (1, 3, 1): f(3):= (1, 3, 3, 3, 1):
A:= [1]:
for i from 1 to 5 do A:= map(f, A) od:
A;
g(1):= (0, 0, 1, 0); g(3):= (0, 0, 1, 0, 1, 0, 1, 0);
map(g, A):

Mathematica

SubstitutionSystem[{0 -> {0, 0, 1, 0}, 1 -> {1, 0, 1, 0}}, 0, 4] // Last (* Jean-François Alcover, Apr 06 2020 *)

Crossrefs

Cf. A036554, A072939, A080426, A284948, A285384 (complement), A297469, A298468, A328190, A328196.

Sequence in context: A284817 A309766 A354033 * A358227 A284524 A226474

Adjacent sequences: A328976 A328977 A328978 * A328980 A328981 A328982

Keyword

nonn

Author

N. J. A. Sloane, Nov 04 2019. Extensively revised Nov 05 2019 thanks to comments from R. J. Mathar.

Status

approved