A301848 Number of states generated by morphism during inflation stage of paper-folding sequence.

1, 3, 1, 4, 1, 3, 2, 4, 1, 3, 1, 4, 2, 3, 2, 4, 1, 3, 1, 4, 1, 3, 2, 4, 2, 3, 1, 4, 2, 3, 2, 4, 1, 3, 1, 4, 1, 3, 2, 4, 1, 3, 1, 4, 2, 3, 2, 4, 2, 3, 1, 4, 1, 3, 2, 4, 2, 3, 1, 4, 2, 3, 2, 4, 1, 3, 1, 4, 1, 3, 2, 4, 1, 3, 1, 4, 2, 3, 2, 4, 1, 3, 1, 4, 1, 3, 2, 4, 2, 3, 1, 4, 2, 3, 2, 4, 2, 3, 1, 4, 1

Offset: 0

Author

Fred Lunnon, Mar 27 2018

Keywords

• nonn

Comments

a(0), a(1), ... is the fixed point of inflation morphism 1 -> 1 3, 2 -> 2 3, 3 -> 1 4, 4 -> 2 4, starting from state 1;

b(0), b(1), ... is the image of a(n) under encoding morphism 1 -> 0, 2 -> 1, 3 -> 0, 4 -> 1.

The number-wall over the rationals (signed Hankel determinants) is apparently free from zeros.

References

• Jean-Paul Allouche and Jeffrey O. Shallit, Automatic sequences, Cambridge, 2003, sect. 5.1.6.

Links

• W. F. Lunnon, The number-wall algorithm: an LFSR cookbook, Journal of Integer Sequences 4 (2001), no. 1, 01.1.1.

Crossrefs

Cf. A038189, A301849, A301850.

Programs

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+n) - 2*b(n+n-1) + 3; end function;
  nlo := 0; nhi := 32;
  [a(n) : n in [nlo..nhi] ];

Maple

A301848 := proc(n)
    A038189(2*n)-2*A038189(2*n-1)+3 ;
end proc:
seq(A301848(n),n=0..100) ; # R. J. Mathar, Mar 30 2018

Mathematica

b[n_] := If[n == 0, 0, BitGet[n, IntegerExponent[n, 2] + 1]];
a[n_] := b[2n] - 2 b[2n-1] + 3;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 30 2023 *)

Formula

a(n) = b(2n) - 2 b(2n-1) + 3, where b(n) denotes A038189(n).