A351898 Decimal expansion of metallic ratio for N = 14.

1, 4, 0, 7, 1, 0, 6, 7, 8, 1, 1, 8, 6, 5, 4, 7, 5, 2, 4, 4, 0, 0, 8, 4, 4, 3, 6, 2, 1, 0, 4, 8, 4, 9, 0, 3, 9, 2, 8, 4, 8, 3, 5, 9, 3, 7, 6, 8, 8, 4, 7, 4, 0, 3, 6, 5, 8, 8, 3, 3, 9, 8, 6, 8, 9, 9, 5, 3, 6, 6, 2, 3, 9, 2, 3, 1, 0, 5, 3, 5, 1, 9, 4, 2, 5, 1, 9

Offset

2,2

Comments

Decimal expansion of continued fraction [14; 14, 14, 14, ...].

Also largest solution of x^2 - 14 x - 1 = 0.

Essentially the same digit sequence as A010503, A157214, A174968 and A268683.

The metallic ratio's for N = A077444(n) are equal to powers of the silver ratio, i.e., A014166^(2n-1); this constant represents the special case for N = A077444(2).

Links

Table of n, a(n) for n=2..88.

Michael Penn, 7+5√2 is a perfect cube??, YouTube video, 2022.

Wikipedia, Metallic mean.

Formula

Equals 2 + 5*A014176.

Equals A014176^3.

Equals exp(arcsinh(7)). - Amiram Eldar, Jul 04 2023

Example

14.0710678118654752440084436210484903928483593...

Mathematica

RealDigits[7 + 5*Sqrt[2], 10, 100][[1]] (* Amiram Eldar, Feb 24 2022 *)

Prog

(PARI) (1+sqrt(2))^3

Crossrefs

Cf. A010503, A077444, A157214, A174968, A268683.

Metallic ratios: A001622 (N=1), A014176 (N=2), A098316 (N=3), A098317 (N=4), A098318 (N=5), A176398 (N=6), A176439 (N=7), A176458 (N=8), A176522 (N=9), A176537 (N=10), A244593 (N=11).

Sequence in context: A052400 A354450 A199071 * A157698 A342360 A251967

Adjacent sequences: A351895 A351896 A351897 * A351899 A351900 A351901

Keyword

nonn,cons

Author

A.H.M. Smeets, Feb 24 2022

Status

approved