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

Author

Stanislav Sykora, May 29 2015

Keywords

• nonn
• cons

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

Examples

			0.69320546462379732043436370422413868794102175016921901339955586752...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..2000
• Index entries for algebraic numbers, degree 8

Crossrefs

Cf. A156548, A156590.

Programs

Mathematica

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

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

Formula

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