A040056 Continued fraction for sqrt(65).

8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• easy
• changed

Examples

			8.06225774829854965236661... = 8 + 1/(16 + 1/(16 + 1/(16 + 1/(16 + ...)))).
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• G. Xiao, Contfrac
• Index entries for continued fractions for constants
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A010517 (decimal expansion), A041112/A041113 (convergents), A248289 (Egyptian fraction).

Cf. A040000, A040002, A040012.

Programs

Maple

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

Mathematica

ContinuedFraction[Sqrt[65],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{8},120,{16}] (* Harvey P. Dale, Nov 27 2020 *)

PARI

{ allocatemem(932245000); default(realprecision, 49000); x=contfrac(sqrt(65)); for (n=0, 20000, write("b040056.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 07 2009

Formula

From Elmo R. Oliveira, Feb 10 2024: (Start)

a(n) = 16 = A010855(n) for n >= 1.

G.f.: 8*(1+x)/(1-x).

E.g.f.: 16*exp(x) - 8.

a(n) = 8*A040000(n) = 4*A040002(n) = 2*A040012(n). (End)