A353920 Decimal expansion of the first positive real root of ((1 - sqrt(5))*((1 + sqrt(5)) /2)^x - (1 + sqrt(5))*((1 - sqrt(5))/2)^x)/(2*sqrt(5)).

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

Offset: 0

Author

Peter Luschny, May 11 2022

Keywords

• nonn
• cons

Comments

The constant is the abscissa of the first intercept point of the row functions for x > 0 of the generalized Fibonacci function A353595, see illustration.

Examples

			0.816197640307044395086030989848733265742877280134657182905...
		

Links

• Peter Luschny, Illustration: Intersection point of A353595.

Crossrefs

Cf. A001622, A353595, A089260.

Programs

Maple

sphi := x -> ((1-sqrt(5))*((1+sqrt(5))/2)^x - (1 + sqrt(5))*((1 - sqrt(5))/2)^x)/ (2*sqrt(5)):
Digits := 120: fsolve(Re(sphi(x)) = 0, x, 0.7..0.9, fulldigits)*10^105:
ListTools:-Reverse(convert(floor(%), base, 10));

Mathematica

sphi[x_] := 2 Re[ Exp[-I Pi x / 2] Sin[(x - 1)(Pi / 2 - I ArcCsch[2])]] / Sqrt[5];
x /. FindRoot[Sphi[x], {x, 0.8}, WorkingPrecision -> 120]
RealDigits[%, 10, 105][[1]]

Formula

Equals the first positive real root of 2*exp(-I*Pi*x/2)*sin((x - 1)*(Pi/2 - I * arccsch(2))) / sqrt(5).