A132338 Decimal expansion of 1 - 1/phi.

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: 0

Author

N. J. A. Sloane, Nov 07 2007

Keywords

• cons
• nonn

Comments

Density of 1's in Fibonacci word A003849.

Also decimal expansion of Sum_{n>=1} ((-1)^(n+1))*1/phi^n. - Michel Lagneau, Dec 04 2011

The Lambert series evaluated at this point is 0.8828541617125076... [see André-Jeannin]. - R. J. Mathar, Oct 28 2012

Because this equals 2 - phi, this is an integer in the quadratic number field Q(sqrt(5)). (Note that this is also sqrt(5 - 3*phi).) - Wolfdieter Lang, Jan 08 2018

When m >= 1, the equation m*x^m + (m-1)*x^(m-1) + ... + 2*x^2 + x - 1 = 0 has only one positive root, u(m) (say); then lim_{m->oo} u(m) = (3-sqrt(5))/2 (see Aubonnet). - Bernard Schott, May 12 2019

Cosine of the zenith angle at which a string should be cut so that a ball tied to one of its ends, set moving without friction around a vertical circle with the minimum speed in a uniform gravitational field, will then travel through the fixed center of the circle. - Stefano Spezia, Oct 25 2020

Algebraic number of degree 2 with minimal polynomial x^2 - 3*x + 1. The other root is 1 + phi = A104457. - Wolfdieter Lang, Aug 29 2022

Examples

			0.38196601125010515179541316563436188...
		

References

• F. Aubonnet, D. Guinin and A. Ravelli, Oral, Concours d'entrée des Grandes Ecoles Scientifiques, Exercices résolus, "Crus" 1982-83, Bréal, 1983, Exercice 210, 40-42.

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• R. André-Jeannin, Lambert series and the summation of reciprocals in certain Fibonacci-Lucas-Type sequences, Fib. Quart. 28 (1990) 223-226.
• Yiyan Ni, Myron Hlynka, and Percy H. Brill, Urn Models and Fibonacci Series, arXiv:1806.09150 [math.CO], 2018. See (9) p. 7.
• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A001622, A003849, A094874, A079585, A094214, A104457.

Programs

Mathematica

RealDigits[N[1/GoldenRatio^2,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)
RealDigits[1-1/GoldenRatio,10,120][[1]] (* Harvey P. Dale, Mar 30 2024 *)

PARI

(3-sqrt(5))/2 \\ Michel Marcus, Oct 26 2020

Formula

Equals 1 - 1/phi = 2 - phi, with phi from A001622.

Equals A094874 - 1, or A079585 - 2, or the square of A094214.

Equals (5-sqrt(5))^2/20 = 1/phi^2 = 1/A104457. - Joost Gielen, Sep 28 2013 [corrected by Joerg Arndt, Sep 29 2013]

Equals (3-sqrt(5))/2. - Bernard Schott, May 12 2019

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