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%I #21 Apr 20 2024 22:14:18
%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]]
%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,changed
%O 1,1
%A _Klaus Brockhaus_, Apr 19 2010
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