A089260 Decimal expansion of -x, the largest negative real root of the equation Fibonacci(x) = 0.

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

Offset: 0

Author

T. D. Noe, Oct 27 2003

Keywords

• cons
• nonn

Comments

For a complex number z, the Fibonacci function is defined as (phi^z - cos(z*Pi) / phi^z) / sqrt(5), where phi is the golden ratio (1 + sqrt(5))/2. There are zeros at z = 0 and an infinite number of negative numbers that approach n + 0.5, for all negative integers n.

Examples

			0.18380235969295560491396901015126673425712271986534281709496082770...
		

Links

• Eric Weisstein's World of Mathematics, Fibonacci Number

Crossrefs

Cf. A089261 (continued fraction), A353920.

Programs

Maple

sphi := x -> ((1/2 - sqrt(5)/2)^x - (1/2 + sqrt(5)/2)^x)/sqrt(5):
Digits := 120: fsolve(Re(sphi(x)) = 0, x, -0.3..-0.1, fulldigits)*10^105:
ListTools:-Reverse(convert(floor(-%), base, 10)); # Peter Luschny, May 11 2022

Mathematica

RealDigits[ -x/.FindRoot[Fibonacci[x]==0, {x, -0.2}, WorkingPrecision->100]][[1]]
(* Or: *)
Sphi[x_] := 2 Im[Sin[(x/2) (Pi - 2 I ArcCsch[2])] / I^x];
x /. FindRoot[Sphi[x], {x, -0.2}, WorkingPrecision -> 120]
RealDigits[%, 10, 105][[1]] (* Peter Luschny, May 11 2022 *)

Formula

Equals the largest negative real root of 2*Im(sin((x/2)*(Pi - 2*i*arccsch(2))) / i^x). - Peter Luschny, May 11 2022