A176439 Decimal expansion of (7+sqrt(53))/2.

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

Offset

1,1

Comments

Continued fraction expansion of (7+sqrt(53))/2 is A010727.

This is the shape of a 7-extension rectangle; see A188640 for definitions. [From Clark Kimberling, Apr 09 2011]

c^n = c * A054413(n-1) + A054413(n-2), where c = (7+sqrt(53))/2. - Gary W. Adamson, Apr 14 2024

Links

Table of n, a(n) for n=1..105.

Wikipedia, Metallic mean

Index entries for algebraic numbers, degree 2

Formula

Equals lim_{n->oo} S(n, sqrt(53))/S(n-1, sqrt(53)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

Positive solution of x^2 - 7*x - 1 = 0. - Hugo Pfoertner, Apr 14 2024

Example

(7+sqrt(53))/2 = 7.14005494464025913554...

Mathematica

r=7; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[(7+Sqrt[53])/2, 10, 120][[1]] (* Harvey P. Dale, Nov 03 2024 *)

Prog

(PARI) (7+sqrt(53))/2 \\ Charles R Greathouse IV, Jul 24 2013

Crossrefs

Cf. A010506 (decimal expansion of sqrt(53)), A010727 (all 7's sequence).

Cf. A049310.

Sequence in context: A089562 A104621 A331900 * A048834 A010504 A242132

Adjacent sequences: A176436 A176437 A176438 * A176440 A176441 A176442

Keyword

nonn,cons,easy

Author

Klaus Brockhaus, Apr 19 2010

Status

approved