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%I #52 Jun 23 2024 04:22:34
%S 1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,0,
%T 0,1,1,0,1,1,0,0,1,1,0,0,0,0,1,1,0,0,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,
%U 0,0,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0
%N Infinite word formed by expanding terms of A372257 by the map h : 0 -> {1,0,1}, 1 -> {1,0,0,1}, 2 -> {1,0,0,0,1}, 3 -> {1,0,0,0,0,1}.
%C This is word "y" of Berstel et al.
%C They construct this sequence (and A372258 "z") as an example infinite word of 2 symbols which is uniformly recurrent, has subwords closed under reversal, but only has a finite set of palindrome subwords.
%C There are 24 palindrome subwords and they can be found using Berstel et al's proof that it's enough to look in all h(T) where T is a length 3 subword occurring in A372257.
%C Blocks of 18 terms a(18*i,...,18*i+17) can be calculated from blocks of 4 terms A372257(4*i,...,4*i+3) since the latter are 0,1 or 1,0 then 2,3 or 3,2 (by the construction there), and so always expand to 18 terms under h.
%C This is a morphic sequence, meaning a symbol to symbol mapping of the fixed point of a morphism (for instance a uniform morphism based on the bits of n and how they result in remainder n mod 18 and bits of the corresponding quotient).
%H Kevin Ryde, <a href="/A369989/b369989.txt">Table of n, a(n) for n = 0..10000</a>
%H Jean Berstel, Luc Boasson, Olivier Carton, and Isabelle Fagnot, <a href="https://arxiv.org/abs/0903.2382">Infinite words without palindrome</a>, arXiv:0903.2382 [cs.DM], 2009, section 3 word "y".
%H Kevin Ryde, <a href="/A369989/a369989.gp.txt">PARI/GP Code</a>
%F With quotient q = floor(n/9) and remainder r = n mod 9:
%F a(n) = 1 - A014707(q) if r = 2;
%F a(n) = A014707(q) if r = 4;
%F a(n) = 1 if q even and r in {0, 3, 6, 7}, or q odd and r in {3, 8};
%F a(n) = 0 otherwise.
%e Expansion of the terms of A372257 by h begins:
%e A372257 = 0, 1, 2, ...
%e this sequence = 1,0,1, 1,0,0,1, 1,0,0,0,1, ...
%o (PARI) \\ See links.
%Y Cf. A014707, A372257, A372258.
%K nonn,easy
%O 0
%A _Kevin Ryde_, May 14 2024
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