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A104457 Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2. 28
2, 6, 1, 8, 0, 3, 3, 9, 8, 8, 7, 4, 9, 8, 9, 4, 8, 4, 8, 2, 0, 4, 5, 8, 6, 8, 3, 4, 3, 6, 5, 6, 3, 8, 1, 1, 7, 7, 2, 0, 3, 0, 9, 1, 7, 9, 8, 0, 5, 7, 6, 2, 8, 6, 2, 1, 3, 5, 4, 4, 8, 6, 2, 2, 7, 0, 5, 2, 6, 0, 4, 6, 2, 8, 1, 8, 9, 0, 2, 4, 4, 9, 7, 0, 7, 2, 0, 7, 2, 0, 4, 1, 8, 9, 3, 9, 1, 1, 3, 7, 4, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Only first term differs from the decimal expansion of Phi.

Zelo extends work of D. Roy by showing that the square of the golden ratio is the optimal exponent of approximation by algebraic numbers of degree 4 with bounded denominator and trace. [Jonathan Vos Post, Mar 02 2009] (cf. last sentence in the Zelo reference. - Joerg Arndt, Jan 04 2014)

Hawkes asks: "What two numbers are those whose product, difference of their squares, and the ratio or quotient of their cubes, are all equal to each other?". - Charles R Greathouse IV, Dec 11 2012

This is the case n=10 in (Gamma(1/n)/Gamma(3/n))*(Gamma((n-1)/n)/Gamma((n-3)/n)) = 1+2*cos(2*Pi/n). - Bruno Berselli, Dec 14 2012

REFERENCES

Damien Roy. Diophantine Approximation in Small Degree. Centre de Recherches Mathematiques. CRM Proceedings and Lecture Notes. Volume 36 (2004), 269-285.

LINKS

Ivan Panchenko, Table of n, a(n) for n = 1..1000

M. Berg, Phi, the golden ratio (to 4599 decimal places) and Fibonacci numbers, Fibonacci Quarterly, 4 (1966), 157-162.

John Hawkes et al., Question 1029, The Mathematical Questions Proposed in the Ladies' Diary (1817), p. 339. Originally published 1798 and answered in 1799.

Casey Mongoven, Phi^2 number 1; electronic music created using Phi^2.

Damien Roy, Diophantine Approximation in Small Degree, arXiv:math/0303150 [math.NT], 2003.

Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions

Eric Weisstein's World of Mathematics, Chromatic Polynomial

Dmitrij Zelo, Simultaneous Approximation to Real and p-adic Numbers, arXiv:0903.0086 [math.NT], 2009.

FORMULA

Equals 2+A094214 = 1+A001622. - R. J. Mathar, May 19 2008

Satisfies these three equations: x-sqrt(x)-1 = 0; x-1/sqrt(x)-2 = 0; x^2-3*x+1 = 0. - Richard R. Forberg, Oct 11 2014

EXAMPLE

2.618033988...

MATHEMATICA

RealDigits[N[GoldenRatio+1, 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

PROG

(PARI) (3+sqrt(5))/2 \\ Charles R Greathouse IV, Aug 21 2012

CROSSREFS

Cf. A001622.

Sequence in context: A021386 A201936 A019679 * A155832 A136764 A136765

Adjacent sequences:  A104454 A104455 A104456 * A104458 A104459 A104460

KEYWORD

nonn,cons,easy,changed

AUTHOR

Eric W. Weisstein, Mar 08 2005

STATUS

approved

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Last modified October 24 01:06 EDT 2014. Contains 248491 sequences.