

A132338


Decimal expansion of 1  1/phi.


10



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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


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 AndréJeannin].  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 + (m1)*x^(m1) + ... + 2*x^2 + x  1 = 0 has only one positive root, u(m) (say); then lim_{m>oo} u(m) = (3sqrt(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


REFERENCES

F. Aubonnet, D. Guinin and A. Ravelli, Oral, Concours d'entrée des Grandes Ecoles Scientifiques, Exercices résolus, "Crus" 198283, Bréal, 1983, Exercice 210, 4042.


LINKS

Ivan Panchenko, Table of n, a(n) for n = 0..1000
R. AndréJeannin, Lambert series and the summation of reciprocals in certain FibonacciLucasType sequences, Fib. Quart. 28 (1990) 223226.


FORMULA

Equals 1  1/phi = 2  phi, with phi from A001622.
Equals A094874  1, or A079585  2, or the square of A094214.
Equals (5sqrt(5))^2/20 = 1/phi^2 = 1/A104457.  Joost Gielen, Sep 28 2013 [corrected by Joerg Arndt, Sep 29 2013]
Equals (3sqrt(5))/2.  Bernard Schott, May 12 2019


EXAMPLE

0.38196601125010515179541316563436188...


MATHEMATICA

RealDigits[N[1/GoldenRatio^2, 200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)


PROG

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


CROSSREFS

Cf. A001622, A003849, A094874, A079585, A094214, A104457.
Sequence in context: A016622 A143623 A094874 * A132702 A197725 A288875
Adjacent sequences: A132335 A132336 A132337 * A132339 A132340 A132341


KEYWORD

cons,nonn


AUTHOR

N. J. A. Sloane, Nov 07 2007


STATUS

approved



