A098318 Decimal expansion of [5, 5, ...] = (5 + sqrt(29))/2.

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, 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, 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

Offset: 1

Author

Eric W. Weisstein, Sep 02 2004

Keywords

• nonn
• cons
• easy

Comments

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

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

Examples

			5.19258240356725201562535524577016477814756...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• A. Stakhov and Samuil Aranson, Hyperbolic Fibonacci and Lucas functions, Golden Fibonacci Goniometry, Bodnar's Geometry, ..., Appl. Math. 2 (1) (2011) 74-84.
• Wikipedia, Metallic mean
• Index entries for algebraic numbers, degree 2

Crossrefs

Cf. A001622, A014176, A098316, A098317, A010716 (continued fraction).

Cf. A052918, A049310.

Programs

Magma

SetDefaultRealField(RealField(100)); (5 + Sqrt(29))/2; // G. C. Greubel, Jun 30 2019

Mathematica

r=5; t=(r+(4+r^2)^(1/2))/2; FullSimplify[t]
N[t,130]
RealDigits[N[t,130]][[1]]
ContinuedFraction[t,120] (* Clark Kimberling, Apr 09 2011 *)

PARI

(5 + sqrt(29))/2 \\ Charles R Greathouse IV, Jul 24 2013

Sage

numerical_approx((5+sqrt(29))/2, digits=100) # G. C. Greubel, Jun 30 2019

Formula

5 plus the constant in A085551. - R. J. Mathar, Sep 02 2008

c^n = A052918(n-2) + A052918(n-1) * c, where c = (5 + sqrt(29))/2. - Gary W. Adamson, Oct 09 2023

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