A335020 Decimal expansion of (1/phi)^(1/phi), where phi is the golden ratio (1 + sqrt(5))/2 (A001622).

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

Offset: 0

Author

Alois P. Heinz, May 19 2020

Keywords

• nonn
• cons

Comments

The real function f(x) = (1/phi)^(1/phi) x^phi satisfies the differential equation f'(x) = f^(-1)(x): the derivative of f equals the compositional inverse of f.

Examples

			0.7427429446246816413695660476057885141497552527...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Dr Peyam, Derivative equals inverse, youtube video (2018)

Crossrefs

Cf. A001622, A094214, A185261.

Programs

Maple

g:= (phi-> (1/phi)^(1/phi))((1+sqrt(5))/2):
evalf(g, 140);

Mathematica

RealDigits[(1/GoldenRatio)^(1/GoldenRatio), 10, 100][[1]] (* Amiram Eldar, May 21 2020 *)

PARI

my(x=(sqrt(5)-1)/2); x^x \\ Michel Marcus, May 21 2020

Formula

Equals (phi-1)^(phi-1), with phi = (1 + sqrt(5))/2.

Equals A094214^A094214.