A391814 Binary expansion of the constant x where a(n) = 2 - A391813(n-1) for n >= 1, and A391813 is the continued fraction of x starting with A391813(0) = 1.

1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0

Offset

1

Comments

This is an example of a constant which has a simple continued fraction expansion with the same parity as its binary expansion.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..5000

Example

x = 1.3867505068375175679941682578045223324514... (A391812).
This binary expansion of x begins (offset 1)
[1,0,1,1,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,0,0,1,
  1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,0,0,0,1,1,1,0,0,1,0,
  1,0,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,1,1,0,0,
  1,0,0,0,0,1,0,1,1,1,1,1,1,1,1,0,0,1,1,0,1,1,1,0,0,
  0,0,1,0,0,0,0,1,0,1,1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,
  0,1,0,0,0,0,1,1,1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,
  0,0,0,1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,1,1,
  0,0,1,0,1,1,1,0,0,1,1,1,0,1,1,1,0,1,1,0,0,1,1,0,0, ...]
Compare with the continued fraction of x (offset 0)
A391813 = [1;2,1,1,2,2,2,1,1,2,2,2,2,2,2,1,2,2,2,2,1,2,1,2,2,1,
  1,2,2,1,2,1,2,1,2,2,1,2,1,2,2,2,2,2,1,1,1,2,2,1,2,
  1,2,1,2,2,1,2,1,2,1,1,2,1,2,1,2,2,2,1,2,1,1,1,2,2,
  1,2,2,2,2,1,2,1,1,1,1,1,1,1,1,2,2,1,1,2,1,1,1,2,2,
  2,2,1,2,2,2,2,1,2,1,1,1,1,2,1,1,1,2,2,2,1,2,2,1,2,
  2,1,2,2,2,2,1,1,1,2,1,1,2,1,1,2,2,1,1,1,2,2,2,1,2,
  2,2,2,1,2,1,2,1,1,2,2,1,1,2,1,2,1,2,1,2,1,2,1,1,1,
  2,2,1,2,1,1,1,2,2,1,1,1,2,1,1,1,2,1,1,2,2,1,1,2,2, ...].
to see that a(n) = 2 - A391813(n-1) for n >= 1.

Prog

(PARI) \\ must set appropriate precision and value of N
{N = 100; r = sqrt(2); for(i=1, N, B = binary(r); C2 = vector(#B[2], k, 2 - B[2][k]); C = concat(1, C2); M = contfracpnqn(C); r = M[1, 1]/M[2, 1]*1.); C}

Crossrefs

Cf. A391812 (decimal expansion), A391813 (continued fraction).

Cf. A391872.

Sequence in context: A275973 A218173 A068426 * A267006 A280816 A265246

Adjacent sequences: A391811 A391812 A391813 * A391815 A391816 A391817

Keyword

nonn,base

Author

Paul D. Hanna, Dec 30 2025

Status

approved