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A301850 The Dakota sequence: a sequence with zero-free number-wall over ternary extension fields. 4
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, 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, -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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0
COMMENTS
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;
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).
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.
REFERENCES
Jean-Paul Allouche and Jeffrey O. Shallit, Automatic sequences, Cambridge, 2003.
LINKS
W. F. Lunnon, The number-wall algorithm: an LFSR cookbook, Journal of Integer Sequences 4 (2001), no. 1, 01.1.1.
Fred Lunnon, The Pagoda sequence: a ramble through linear complexity, number walls, D0L sequences, finite state automata, and aperiodic tilings, Electronic Proceedings in Theoretical Computer Science 1 (2009), 130-148.
MATHEMATICA
b[n_] := b[n] = If[n == 0, 0, BitGet[n, IntegerExponent[n, 2] + 1]];
c[n_] := b[2 n] - 2 b[2 n - 1] + 3;
Array[c, 50, 0] /. {1 -> {0, 1}, 2 -> {1, -1}, 3 -> {0, -1}, 4 -> {1, 0}} // Flatten (* Jean-François Alcover, Dec 13 2018 *)
PROG
(Magma)
function b (n)
if n eq 0 then return 0; // alternatively, return 1;
else while IsEven(n) do n := n div 2; end while; end if;
return n div 2 mod 2; end function;
function c (n)
return b(n+n) - 2*b(n+n-1) + 3; end function;
PGF<x> := PolynomialRing(RationalField()); // polynomial in x
function xplusa (n, x)
return [ [x, x+1], [x+1, x-1], [x, x-1], [x+1, x] ]
[c(n div 2)][n mod 2+1];
end function;
function a (n)
return Coefficient(xplusa(n, x), 0); end function;
nlo := 0; nhi := 32;
[a(n) : n in [nlo..nhi] ];
CROSSREFS
Sequence in context: A285515 A190204 A189028 * A189031 A189212 A147781
KEYWORD
sign
AUTHOR
Fred Lunnon, Mar 27 2018
EXTENSIONS
More terms from Jean-François Alcover, Dec 13 2018
STATUS
approved

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Last modified March 3 20:15 EST 2024. Contains 370512 sequences. (Running on oeis4.)