A098317 - OEIS

%I #70 Nov 15 2023 04:56:07

%S 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,

%T 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,

%U 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

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

%C This sequence is also the decimal expansion of ((1+sqrt(5))/2)^3. - _Mohammad K. Azarian_, Apr 14 2008

%C This is the length/width ratio of a 4-extension rectangle; see A188640 for definitions. - _Clark Kimberling_, Apr 10 2011

%C Its continued fraction is [4, 4, ...] (see A010709). - _Robert G. Wilson v_, Apr 10 2011

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

%H G. C. Greubel, <a href="/A098317/b098317.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Metallic_mean">Metallic mean</a>

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>

%F 2 plus the constant in A002163. - _R. J. Mathar_, Sep 02 2008

%F 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

%F 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

%F Equals exp(arcsinh(2)), since arcsinh(x) = log(x+sqrt(x^2+1)). - _Stanislav Sykora_, Nov 01 2013

%F Equals Sum_{n>=1} n/phi^n = phi/(phi-1)^2 = phi^3. - _Richard R. Forberg_, Jun 29 2014

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

%F c^n = A001076(n-1) + c * A001076(n); where c = 2 + sqrt(5). - _Gary W. Adamson_, Oct 09

%F 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

%e 4.23606797749978969640917366873127623544061835961152572427...

%t RealDigits[N[2+Sqrt[5],200]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 21 2011 *)

%o (PARI) sqrt(5)+2 \\ _Charles R Greathouse IV_, Mar 11 2012

%o (Magma) SetDefaultRealField(RealField(100)); 2+Sqrt(5); // _G. C. Greubel_, Jun 30 2019

%o (Sage) numerical_approx(2+sqrt(5), digits=100) # _G. C. Greubel_, Jun 30 2019

%Y Cf. A001622, A014176, A098316, A098318.

%Y Cf. A001076, A049310.

%K nonn,cons,easy

%O 1,1

%A _Eric W. Weisstein_, Sep 02 2004

%E Title expanded to include observation from _Mohammad K. Azarian_ by _Charles R Greathouse IV_, Mar 11 2012