

A098317


Decimal expansion of phi^3 = 2 + sqrt(5).


28



4, 2, 3, 6, 0, 6, 7, 9, 7, 7, 4, 9, 9, 7, 8, 9, 6, 9, 6, 4, 0, 9, 1, 7, 3, 6, 6, 8, 7, 3, 1, 2, 7, 6, 2, 3, 5, 4, 4, 0, 6, 1, 8, 3, 5, 9, 6, 1, 1, 5, 2, 5, 7, 2, 4, 2, 7, 0, 8, 9, 7, 2, 4, 5, 4, 1, 0, 5, 2, 0, 9, 2, 5, 6, 3, 7, 8, 0, 4, 8, 9, 9, 4, 1, 4, 4, 1, 4, 4, 0, 8, 3, 7, 8, 7, 8, 2, 2, 7, 4, 9, 6
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

This sequence is also the decimal expansion of ((1+sqrt(5))/2)^3.  Mohammad K. Azarian, Apr 14 2008
This is the length/width ratio of a 4extension rectangle; see A188640 for definitions. [Clark Kimberling, Apr 10 2011]
Its continued fraction is [4, 4, ...] (see A040002).  Robert G. Wilson v, Apr 10 2011
lim n > infinity F(n+3)/F(n) = lim n > infinity 1 + 2*F(n+1)/F(n) = 2 + sqrt(5), with F(n) = A000045(n). [Arkadiusz Wesolowski, Mar 11 2012]
Sum{n = 1 to infinity, n/phi^n} = phi/(phi1)^2 = phi^3.  Richard R. Forberg, Jun 29 2014


REFERENCES

Alexey Stakhov, The mathematics of harmony: from Euclid to contemporary mathematics and computer science, World Scientific, Singapore, 2009, p. 657.


LINKS

Table of n, a(n) for n=1..102.


FORMULA

2 plus the constant in A002163. [R. J. Mathar, Sep 02 2008]
Equals 3 + 4*sin(Pi/10) = 1 + 4*cos(Pi/5) = 1 + 4*sin(3*Pi/10) = 3 + 4*cos(2*Pi/5) = 1 + csc(Pi/10). [Arkadiusz Wesolowski, Mar 11 2012]
Equals exp(arcsinh(2)), since arcsinh(x) = log(x+sqrt(x^2+1)).  Stanislav Sykora, Nov 01 2013


EXAMPLE

4.23606797...


MATHEMATICA

RealDigits[N[2+Sqrt[5], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011*)


PROG

(PARI) sqrt(5)+2 \\ Charles R Greathouse IV, Mar 11, 2012


CROSSREFS

Cf. A001622, A014176, A098316, A098318.
Sequence in context: A226939 A123403 A243961 * A095185 A128009 A072425
Adjacent sequences: A098314 A098315 A098316 * A098318 A098319 A098320


KEYWORD

nonn,cons,easy


AUTHOR

Eric W. Weisstein, Sep 02, 2004


EXTENSIONS

Expanded title to include observation of Mohammad K. Azarian, Charles R Greathouse IV, Mar 11, 2012


STATUS

approved



