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

%I A089260 #12 Feb 16 2025 08:32:51
%S A089260 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,
%T A089260 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,
%U A089260 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
%N A089260 Decimal expansion of -x, the largest negative real root of the equation Fibonacci(x) = 0.
%C A089260 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.
%H A089260 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%F A089260 Equals the largest negative real root of 2*Im(sin((x/2)*(Pi - 2*i*arccsch(2))) / i^x). - _Peter Luschny_, May 11 2022
%e A089260 0.18380235969295560491396901015126673425712271986534281709496082770...
%p A089260 sphi := x -> ((1/2 - sqrt(5)/2)^x - (1/2 + sqrt(5)/2)^x)/sqrt(5):
%p A089260 Digits := 120: fsolve(Re(sphi(x)) = 0, x, -0.3..-0.1, fulldigits)*10^105:
%p A089260 ListTools:-Reverse(convert(floor(-%), base, 10)); # _Peter Luschny_, May 11 2022
%t A089260 RealDigits[ -x/.FindRoot[Fibonacci[x]==0, {x, -0.2}, WorkingPrecision->100]][[1]]
%t A089260 (* Or: *)
%t A089260 Sphi[x_] := 2 Im[Sin[(x/2) (Pi - 2 I ArcCsch[2])] / I^x];
%t A089260 x /. FindRoot[Sphi[x], {x, -0.2}, WorkingPrecision -> 120]
%t A089260 RealDigits[%, 10, 105][[1]] (* _Peter Luschny_, May 11 2022 *)
%Y A089260 Cf. A089261 (continued fraction), A353920.
%K A089260 cons,nonn
%O A089260 0,2
%A A089260 _T. D. Noe_, Oct 27 2003