%I #37 Oct 02 2024 08:35:01

%S 7,1,2,1,14,1,2,1,14,1,2,1,14,1,2,1,14,1,2,1,14,1,2,1,14,1,2,1,14,1,2,

%T 1,14,1,2,1,14,1,2,1,14,1,2,1,14,1,2,1,14,1,2,1,14,1,2,1,14,1,2,1,14,

%U 1,2,1,14,1,2,1,14,1,2,1,14

%N Continued fraction for sqrt(60).

%H Harry J. Smith, <a href="/A040052/b040052.txt">Table of n, a(n) for n = 0..20000</a>

%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>.

%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).

%F From _Bruno Berselli_, Mar 07 2011: (Start)

%F G.f.: (7 + x + 2*x^2 + x^3 + 7*x^4)/(1-x^4).

%F a(n) = (6*(-i)^n + 6*i^n + 7*(-1)^n + 9)/2 - 7*A000007(n), where i is the imaginary unit. (End)

%F From _Amiram Eldar_, Nov 13 2023: (Start)

%F Multiplicative with a(2) = 2, a(2^e) = 14 for e >= 2, and a(p^e) = 1 for an odd prime p.

%F Dirichlet g.f.: zeta(s) * (1 + 1/2^s + 3/4^(s-1)). (End)

%e 7.74596669241483377035853079... = 7 + 1/(1 + 1/(2 + 1/(1 + 1/(14 + ...)))). - _Harry J. Smith_, Jun 07 2009

%p Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

%t ContinuedFraction[Sqrt[60],300] (* _Vladimir Joseph Stephan Orlovsky_, Mar 07 2011 *)

%t PadRight[{7},120,{14,1,2,1}] (* _Harvey P. Dale_, Aug 07 2019 *)

%o (PARI) { allocatemem(932245000); default(realprecision, 19000); x=contfrac(sqrt(60)); for (n=0, 20000, write("b040052.txt", n, " ", x[n+1])); } \\ _Harry J. Smith_, Jun 07 2009

%o (Magma) [7] cat &cat[ [1, 2, 1, 14]: n in [1..18]]; // _Bruno Berselli_, Mar 07 2011

%Y Cf. A000007, A010513 (decimal expansion), A248285 (Egyptian fractions).

%K nonn,cofr,easy,mult

%O 0,1

%A _N. J. A. Sloane_