A098317 - OEIS

A098317

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

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 4-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 10 2011

Its continued fraction is [4, 4, ...] (see A010709). - Robert G. Wilson v, Apr 10 2011

REFERENCES

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

Kunle Adegoke, Some Infinite Product Identities Involving Fibonacci and Lucas Numbers, The Fibonacci Quarterly, Vol. 55, No. 4 (2017), pp. 343-351.

Wikipedia, Metallic mean.

Index entries for algebraic numbers, degree 2.

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 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

Equals exp(arcsinh(2)), since arcsinh(x) = log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013

Equals Sum_{n>=1} n/phi^n = phi/(phi-1)^2 = phi^3. - Richard R. Forberg, Jun 29 2014

Equals 1 + 2*phi, with phi = A001622, an integer in the quadratic number field Q(sqrt(5)). - Wolfdieter Lang, Dec 10 2022

c^n = A001076(n-1) + c * A001076(n); where c = 2 + sqrt(5). - Gary W. Adamson, Oct 09 2023

Equals lim_{n -> infinity} = S(n, 2*(-1 + 2*phi))/S(n-1, 2*(-1 + 2*phi)), with the S-Chebyshev polynomials (see A049310). See also the above limit formula with Fibonacci numbers. - Wolfdieter Lang, Nov 15 2023

From Amiram Eldar, Jan 03 2026: (Start)

Formulas from Adegoke (2017):

Equals Product_{k>=1} (sqrt(5)*Fibonacci(2*k)+1)/(sqrt(5)*Fibonacci(2*k)-1), where Fibonacci(n) = A000045(n).

Equals Product_{k>=1} (Lucas(2*k)-(-1)^k*sqrt(5))/(Lucas(2*k)+(-1)^k*sqrt(5)), where Lucas(n) = A000032(n). (End)

EXAMPLE

4.23606797749978969640917366873127623544061835961152572427...

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

(Magma) SetDefaultRealField(RealField(100)); 2+Sqrt(5); // G. C. Greubel, Jun 30 2019

(SageMath) numerical_approx(2+sqrt(5), digits=100) # G. C. Greubel, Jun 30 2019

CROSSREFS

Cf. A000032, A000045, A001622, A014176, A098316, A098318.

Cf. A001076, A049310.

Sequence in context: A276957 A275847 A243961 * A371325 A095185 A128009

Adjacent sequences: A098314 A098315 A098316 * A098318 A098319 A098320

KEYWORD

nonn,cons,easy

AUTHOR

Eric W. Weisstein, Sep 02 2004

EXTENSIONS

Title expanded to include observation from Mohammad K. Azarian by Charles R Greathouse IV, Mar 11 2012

STATUS

approved