%I #37 Nov 15 2023 05:09:54
%S 5,1,9,2,5,8,2,4,0,3,5,6,7,2,5,2,0,1,5,6,2,5,3,5,5,2,4,5,7,7,0,1,6,4,
%T 7,7,8,1,4,7,5,6,0,0,8,0,8,2,2,3,9,4,4,1,8,8,4,0,1,9,4,3,3,5,0,0,8,3,
%U 2,2,9,8,1,4,1,3,8,2,9,3,4,6,4,3,8,3,1,6,8,9,0,8,3,9,9,1,7,7,4,2,2,0
%N Decimal expansion of [5, 5, ...] = (5 + sqrt(29))/2.
%C The "metallic" constants A001622, A014176 etc. are defined inserting a = 1, 2, 3, 4, ... into (a+sqrt(a^2+4))/2. [Stakhov & Aranson] - _R. J. Mathar_, Feb 14 2011
%C This is the length/width ratio of a 5-extension rectangle; see A188640 where the metallic constants are defined for rational numbers. - _Clark Kimberling_, Apr 09 2011
%H G. C. Greubel, <a href="/A098318/b098318.txt">Table of n, a(n) for n = 1..10000</a>
%H A. Stakhov and Samuil Aranson, <a href="http://dx.doi.org/10.4236/am.2011.21009">Hyperbolic Fibonacci and Lucas functions, Golden Fibonacci Goniometry, Bodnar's Geometry, ...</a>, Appl. Math. 2 (1) (2011) 74-84.
%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 5 plus the constant in A085551. - _R. J. Mathar_, Sep 02 2008
%F c^n = A052918(n-2) + A052918(n-1) * c, where c = (5 + sqrt(29))/2. - _Gary W. Adamson_, Oct 09 2023
%F Equals lim_{n->infinity} S(n, sqrt(29))/ S(n-1, sqrt(29)), with the S-Chebyshev polynomials (see A049310). - _Wolfdieter Lang_, Nov 15 2023
%e 5.19258240356725201562535524577016477814756...
%t r=5; t=(r+(4+r^2)^(1/2))/2; FullSimplify[t]
%t N[t,130]
%t RealDigits[N[t,130]][[1]]
%t ContinuedFraction[t,120] (* _Clark Kimberling_, Apr 09 2011 *)
%o (PARI) (5 + sqrt(29))/2 \\ _Charles R Greathouse IV_, Jul 24 2013
%o (Magma) SetDefaultRealField(RealField(100)); (5 + Sqrt(29))/2; // _G. C. Greubel_, Jun 30 2019
%o (Sage) numerical_approx((5+sqrt(29))/2, digits=100) # _G. C. Greubel_, Jun 30 2019
%Y Cf. A001622, A014176, A098316, A098317, A010716 (continued fraction).
%Y Cf. A052918, A049310.
%K nonn,cons,easy
%O 1,1
%A _Eric W. Weisstein_, Sep 02 2004