A301849 - OEIS

%I #29 Oct 10 2024 15:59:04

%S -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,

%T 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,

%U 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

%N The Pagoda sequence: a sequence with isolated zeros in its number wall over finite 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 -> -1,-1,0,+1, 2 -> 0,-1,-1,+1, 3 -> 0,-1,+1,+1, 4 -> +1,-1,0,+1;

%C where c(n) denotes A301848(n).

%C 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.

%D Jean-Paul Allouche and Jeffrey O. Shallit, Automatic sequences, Cambridge, 2003.

%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.

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

%F a(n) = a(-n) for all n in Z except |n|=1. a(-1) = -1. - _Michael Somos_, Oct 05 2024

%t b[n_] := b[n] = If[n == 0, 0, BitGet[n, IntegerExponent[n, 2] + 1]];

%t a[n_] := b[n+1] - b[n-1];

%t Array[a, 100, 0] (* _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 a (n)

%o   return b(n+1) - b(n-1); end function;

%o   nlo := 0; nhi := 32;

%o   [a(n) : n in [nlo..nhi] ];

%Y Cf. A038189, A301848, A301850.

%K sign

%O 0

%A _Fred Lunnon_, Mar 27 2018

%E More terms from _Jean-François Alcover_, Dec 13 2018