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A301850 The Dakota sequence: a sequence with zero-free number-wall over ternary extension fields. 3
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

Table of n, a(n) for n=0..99.

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

Cf. A038189, A301848, A301849.

Sequence in context: A285515 A190204 A189028 * A189031 A189212 A147781

Adjacent sequences:  A301847 A301848 A301849 * A301851 A301852 A301853

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 August 17 23:13 EDT 2022. Contains 356204 sequences. (Running on oeis4.)