A094874 Decimal expansion of (5-sqrt(5))/2.

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

Offset: 1

Author

N. J. A. Sloane, Jun 14 2004

Keywords

• cons
• nonn
• easy

Comments

Also the limiting ratio of Lucas(n)/Fibonacci(n+1), or Fibonacci(n-1)/Fibonacci(n+1) + 1. - Alexander Adamchuk, Oct 10 2007

Examples

			1.38196601125010515179541316563436188...
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..1000
• Paul Cooijmans, Odds.
• Yiyan Ni, Myron Hlynka, and Percy H. Brill, Urn Models and Fibonacci Series, arXiv:1806.09150 [math.CO], 2018. See (9) p. 7.
• Jonathan Sondow, Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers, arXiv:1106.4246 [math.NT], 2011; Diophantine Analysis and Related Fields 2011 - AIP Conference Proceedings, vol. 1385, pp. 97-100.
• Index entries for algebraic numbers, degree 2.

Crossrefs

Equals A079585-1.

Cf. A000032, A000045, A081012, A182007, A187426, A187798, A192223, A242671, A244847.

Programs

Mathematica

RealDigits[5/2 - Sqrt[5]/2, 10, 100][[1]] (* Alonso del Arte, Jun 26 2018 *)

PARI

(5-sqrt(5))/2 \\ Charles R Greathouse IV, Jun 26 2011

Formula

Equals (2-phi)*(2+phi) = 2 - 1/phi = 3 - phi = (5-sqrt(5))/2 = (2*sin(Pi/5))^2, where phi is the golden ratio (A001622).

Equals Product_{n > 0} (1 + 1/A192223(n)). - Charles R Greathouse IV, Jun 26 2011

Equals 1 + Sum_{k >= 2} (-1)^k/(Fibonacci(k)*Fibonacci(k+1)). See Ni et al. - Michel Marcus, Jun 26 2018; corrected by Michel Marcus, Mar 11 2024

Equals Sum_{k>=0} binomial(2*k,k)/((k+1) * 5^k). - Amiram Eldar, Aug 03 2020

From Amiram Eldar, Nov 28 2024: (Start)

Equals 5*A244847 = 2*A187798 = 1/A242671 = A182007^2 = sqrt(A187426).

Equals Product_{k>=1} (1 + 1/A081012(k)). (End)