%I #23 Nov 03 2024 17:58:03

%S 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,

%T 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,

%U 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

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

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

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

%C c^n = c * A054413(n-1) + A054413(n-2), where c = (7+sqrt(53))/2. - _Gary W. Adamson_, Apr 14 2024

%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 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

%F Positive solution of x^2 - 7*x - 1 = 0. - _Hugo Pfoertner_, Apr 14 2024

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

%t r=7; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]

%t N[t, 130]

%t RealDigits[N[t, 130]][[1]]

%t RealDigits[(7+Sqrt[53])/2,10,120][[1]] (* _Harvey P. Dale_, Nov 03 2024 *)

%o (PARI) (7+sqrt(53))/2 \\ _Charles R Greathouse IV_, Jul 24 2013

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

%Y Cf. A049310.

%K nonn,cons,easy

%O 1,1

%A _Klaus Brockhaus_, Apr 19 2010