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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• mult

Examples

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

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).

Crossrefs

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

Programs

Magma

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

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 *)

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

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)