A257945 Decimal expansion of abs(i/(i + i/(i + i/...))) and abs(i/(1 + i/(1 + i/...))), i being the imaginary unit.

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

Offset

0,1

Comments

Set v = A156590 and u = (A156548 - 1). Then the continued fractions evaluate to i/(i + i/(i + i/...)) = (sqrt(4*i - 1) - i)/2 = v + u*i and i/(1 + i/(1 + i/...)) = (sqrt(4*i + 1) - 1)/2 = u + v*i. They can be evaluated either explicitly or as limits of the convergent recursive mappings z -> i/(i + z) and z -> i/(1 + z), respectively, starting, for example, with z = 0.

An algebraic integer of degree 8. - Charles R Greathouse IV, Jun 02 2015

Links

Stanislav Sykora, Table of n, a(n) for n = 0..2000

Index entries for algebraic numbers, degree 8

Formula

Equals sqrt(1 + sqrt(17) - sqrt(2*(1 + sqrt(17))))/2.

Example

0.69320546462379732043436370422413868794102175016921901339955586752...

Mathematica

RealDigits[Sqrt[1 + Sqrt[17] - Sqrt[2*(1 + Sqrt[17])]]/2, 10, 105][[1]] (* Vaclav Kotesovec, Jun 02 2015 *)

Prog

(PARI) sqrt(1+sqrt(17)-sqrt(2*(1+sqrt(17))))/2
(PARI) polrootsreal(x^8-x^6-2*x^4-x^2+1)[3] \\ Charles R Greathouse IV, Jun 02 2015

Crossrefs

Cf. A156548, A156590.

Sequence in context: A022698 A013707 A002162 * A271526 A072365 A239068

Adjacent sequences: A257942 A257943 A257944 * A257946 A257947 A257948

Keyword

nonn,cons

Author

Stanislav Sykora, May 29 2015

Status

approved