A301849 The Pagoda sequence: a sequence with isolated zeros in number-wall over finite fields.

-1, 0, 1, 0, -1, 1, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 0, -1, 1, 1, 0, -1, -1, 1, 1, -1, 0, 1, -1, -1, 0, 1, 0, -1, 1, 1, -1, -1, 0, 1, 1, -1, 0, 1, 0, -1, -1, 1, 0, -1, 1, 1, 0, -1, -1, 1, 1, -1, 0, 1, -1, -1, 0, 1, 0, -1, 1, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 0, -1, 1, 1, 0, -1, -1, 1, 1, -1, 0, 1, 0, -1, -1, 1, 0

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 -> -1,-1,0,+1, 2 -> 0,-1,-1,+1, 3 -> 0,-1,+1,+1, 4 -> +1,-1,0,+1;

where c(n) denotes A301848(n).

The number-walls (signed Hankel determinants) over finite fields with characteristic -1 (mod 4) of this sequence have apparently only isolated zeros, though that has been proved only for p = 3,7.

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.

Formula

a(n) = b(n+1) - b(n-1), where b(n) denotes A038189(n).

Mathematica

b[n_] := b[n] = If[n == 0, 0, BitGet[n, IntegerExponent[n, 2] + 1]];
a[n_] := b[n+1] - b[n-1];
Array[a, 100, 0] (* 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 a (n)
  return b(n+1) - b(n-1); end function;
  nlo := 0; nhi := 32;
  [a(n) : n in [nlo..nhi] ];

Crossrefs

Cf. A038189, A301848, A301850.

Sequence in context: A243566 A266112 A331990 * A074332 A152065 A267870

Adjacent sequences: A301846 A301847 A301848 * A301850 A301851 A301852

Keyword

sign

Author

Fred Lunnon, Mar 27 2018

Extensions

More terms from Jean-François Alcover, Dec 13 2018

Status

approved