A040052 Continued fraction for sqrt(60).

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, 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, 1, 2, 1, 14, 1, 2, 1, 14, 1, 2, 1, 14

Offset

0,1

Links

Harry J. Smith, Table of n, a(n) for n = 0..20000

G. Xiao, Contfrac.

Index entries for continued fractions for constants.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Formula

From Bruno Berselli, Mar 07 2011: (Start)

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

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

From Amiram Eldar, Nov 13 2023: (Start)

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

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

Example

7.74596669241483377035853079... = 7 + 1/(1 + 1/(2 + 1/(1 + 1/(14 + ...)))). - Harry J. Smith, Jun 07 2009

Maple

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

Mathematica

ContinuedFraction[Sqrt[60], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 07 2011 *)
PadRight[{7}, 120, {14, 1, 2, 1}] (* Harvey P. Dale, Aug 07 2019 *)

Prog

(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
(Magma)  [7] cat &cat[ [1, 2, 1, 14]: n in [1..18]]; // Bruno Berselli, Mar 07 2011

Crossrefs

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

Sequence in context: A369980 A090202 A030172 * A134898 A371946 A176440

Adjacent sequences: A040049 A040050 A040051 * A040053 A040054 A040055

Keyword

nonn,cofr,easy,mult

Author

N. J. A. Sloane

Status

approved