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

0,2

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

Table of n, a(n) for n=0..100.

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

Formula

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

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 *)

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

Crossrefs

Cf. A038189, A301849, A301850.

Sequence in context: A094603 A165595 A213181 * A325610 A278536 A143825

Adjacent sequences: A301845 A301846 A301847 * A301849 A301850 A301851

Keyword

nonn

Author

Fred Lunnon, Mar 27 2018

Status

approved