A391812 Decimal expansion of the constant x where each term in the simple continued fraction of x equals 2 minus the respective bit in the binary expansion of x, with an initial term of '1'.

1, 3, 8, 6, 7, 5, 0, 5, 0, 6, 8, 3, 7, 5, 1, 7, 5, 6, 7, 9, 9, 4, 1, 6, 8, 2, 5, 7, 8, 0, 4, 5, 2, 2, 3, 3, 2, 4, 5, 1, 4, 0, 1, 5, 9, 7, 2, 2, 1, 6, 4, 2, 1, 3, 5, 6, 7, 7, 4, 2, 0, 6, 6, 0, 3, 9, 8, 1, 2, 5, 5, 4, 3, 9, 9, 0, 1, 0, 8, 7, 8, 6, 7, 2, 0, 3, 3, 3, 6, 8, 6

Offset

1,2

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..4001

Formula

x = Sum_{n>=0} (2 - A391813(n))/2^n, where A391813 is the continued fraction expansion of x starting with A391813(0) = 1.

Example

The initial 1000 decimal digits of this constant are
1.38675050683751756799416825780452233245140159722164
  21356774206603981255439901087867203336863916309407
  57899137840660613668112023726078855532576022526446
  57274069094450226291382226916624688302773410480774
  08528107291771924747168434839267744464959837965684
  34844363108071461414791343801601691467193986492909
  39765992226906411614541940335977647670287568218652
  43324635079013112790875388908730056063606465927901
  76502668921835447941215239019445212073046662628830
  91648518107909497811947450950868014735949891777369
  21153518771269054774807575088690196100036187087152
  83179978109654303348476505250282858153111623883106
  09057577212108380229606755593006257959388399430382
  16993332840588220509207591636795734634875799774144
  05925374653025828711222676661962541560612472346771
  84344346784202917309771740380429603781087814539742
  41224232716057234655247783691498197013624112147906
  34636658939128725368648795594606125472871539084046
  49372549880232757371561442846766280701915703555203
  49247037356233698060567528364657985051727168391922...
The continued fraction of this constant begins (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, ...].
Compare with the binary expansion of this constant (offset 1)
A391814 = [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, ...]
to see that A391813(n) = 2 - A391814(n+1) for n >= 0.

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.); r}

Crossrefs

Cf. A391813 (continued fraction), A391814 (binary expansion).

Sequence in context: A187061 A363361 A020809 * A152683 A154199 A083700

Adjacent sequences: A391809 A391810 A391811 * A391813 A391814 A391815

Keyword

nonn,cons

Author

Paul D. Hanna, Dec 30 2025

Status

approved