A322258 Decimal expansion of exp(-phi/sqrt(5)), where phi is the golden ratio.

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

Offset: 0

Author

Amiram Eldar, Dec 01 2018

Keywords

• nonn
• cons

Examples

			0.48499980129295802523175132230095248348065996564155...
		

References

• J. Sandor and B. Crstici, Handbook of Number Theory II, Springer, 2004, pp. 54-55, p. 182.

Links

• Don Redmond, Infinite products and Fibonacci numbers, Fib. Quart., Vol. 32, No. 3 (1994), pp. 234-239.
• Index entries for transcendental numbers

Crossrefs

Cf. A000010, A000032, A000045, A001622, A242671, A322259.

Programs

Mathematica

RealDigits[Exp[-GoldenRatio/Sqrt[5]], 10, 120][[1]]

PARI

exp(-(1+1/sqrt(5))/2) \\ Charles R Greathouse IV, Nov 21 2024

Formula

Equals Product_{k>=1} (L(k)/(sqrt(5)*F(k)))^(phi(k)/k), where L(k) and F(k) are the Lucas and Fibonacci numbers, and phi(k) is the Euler totient function.

Equals exp(-A242671).