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%I #19 Nov 01 2023 01:17:40
%S 0,1,0,-1,0,1,1,0,0,1,0,-1,1,-1,1,0,0,1,0,-1,0,1,1,0,1,-1,0,-1,1,-1,1,
%T 0,0,1,0,-1,0,1,1,0,0,1,0,-1,1,-1,1,0,1,-1,0,-1,0,1,1,0,1,-1,0,-1,1,
%U -1,1,0,0,1,0,-1,0,1,1,0,0,1,0,-1,1,-1,1,0,0,1,0,-1,0,1,1,0,1,-1,0,-1,1,-1,1,0,1,-1,0,-1
%N The Dakota sequence: a sequence with zero-free number-wall over ternary extension fields.
%C c(0), c(1), ... is the fixed point of inflation morphism 1 -> 1 3, 2 -> 2 3, 3 -> 1 4, 4 -> 2 4, starting from state 1;
%C a(-1), a(0), ... is the image of c(n) under encoding morphism 1 -> 0,+1; 2 -> +1,-1; 3 -> 0,-1; 4 -> +1,0; where c(n) denotes A301848(n).
%C The number-walls (signed Hankel determinants) over finite fields with characteristic 3 of sequence x + a(n) with x not in F_3 have been proved free of zeros.
%D Jean-Paul Allouche and Jeffrey O. Shallit, Automatic sequences, Cambridge, 2003.
%H Joerg Arndt, <a href="/A301850/b301850.txt">Table of n, a(n) for n = 0..9999</a>
%H W. F. Lunnon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LUNNON/numbwall10.html">The number-wall algorithm: an LFSR cookbook</a>, Journal of Integer Sequences 4 (2001), no. 1, 01.1.1.
%H Fred Lunnon, <a href="https://arxiv.org/abs/0906.3286">The Pagoda sequence: a ramble through linear complexity, number walls, D0L sequences, finite state automata, and aperiodic tilings</a>, Electronic Proceedings in Theoretical Computer Science 1 (2009), 130-148.
%t b[n_] := b[n] = If[n == 0, 0, BitGet[n, IntegerExponent[n, 2] + 1]];
%t c[n_] := b[2 n] - 2 b[2 n - 1] + 3;
%t Array[c, 50, 0] /. {1 -> {0, 1}, 2 -> {1, -1}, 3 -> {0, -1}, 4 -> {1, 0}} // Flatten (* _Jean-François Alcover_, Dec 13 2018 *)
%o (Magma)
%o function b (n)
%o if n eq 0 then return 0; // alternatively, return 1;
%o else while IsEven(n) do n := n div 2; end while; end if;
%o return n div 2 mod 2; end function;
%o function c (n)
%o return b(n+n) - 2*b(n+n-1) + 3; end function;
%o PGF<x> := PolynomialRing(RationalField()); // polynomial in x
%o function xplusa (n, x)
%o return [ [x, x+1], [x+1, x-1], [x, x-1], [x+1, x] ]
%o [c(n div 2)][n mod 2+1];
%o end function;
%o function a (n)
%o return Coefficient(xplusa(n, x), 0); end function;
%o nlo := 0; nhi := 32;
%o [a(n) : n in [nlo..nhi] ];
%Y Cf. A038189, A301848, A301849.
%K sign
%O 0
%A _Fred Lunnon_, Mar 27 2018
%E More terms from _Jean-François Alcover_, Dec 13 2018
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